(Section.3: Basic Graphs and Symmetry).3. SECTION.3: BASIC GRAPHS and SYMMETRY LEARNING OBJECTIVES Know how to graph basic functions. Organize categories of basic graphs and recognize common properties, such as symmetry. Identify which basic functions are even / odd / neither and relate this to symmetry in their graphs. PART A: DISCUSSION We will need to know the basic functions and graphs in this section without resorting to point-plotting. To help us remember them, we will organize them into categories. What are the similarities and differences within and between categories, particularly with respect to shape and symmetry in graphs? (We will revisit symmetry in Section.4 and especially in Section.7.) A power function f has a rule of the form f ( x)= x n, where the exponent or power n is a real number. We will consider graphs of all power functions with integer powers, and some power functions with non-integer powers. In the next few sections, we will manipulate and combine these building blocks to form a wide variety of functions and graphs.
(Section.3: Basic Graphs and Symmetry).3.2 PART B: CONSTANT FUNCTIONS If f ( x)= c, where c is a real number, then f is a constant function. Any real input yields the same output, c. If f ( x)= 3, for example, we have the input-output model and the flat graph of y = 3, a horizontal line, below. PART C: IDENTITY FUNCTIONS If f ( x)= x, then f is an identity function. Its output is identical to its input. 6 f 6 0 f 0 There are technically different identity functions on different domains. The graph of y = x is the line below.
(Section.3: Basic Graphs and Symmetry).3.3 PART D: LINEAR FUNCTIONS If f ( x)= mx + b, where m and b are real numbers, and m 0, then f is a linear function. In Section 0.4, we graphed y = mx + b as a line with slope m and y-intercept b. If f ( x)= 2x, for example, we graph the line with slope 2 and y-intercept. PART E: SQUARING FUNCTION and EVEN FUNCTIONS Let f ( x)= x 2. We will construct a table and graph f. x ( ) Point x f ( x) Point ( ) 0 0 ( 0, 0) ( ) (, ) ( ) 2 4 ( 2, 4) ( ) 3 9 ( 3, 9) f x 0 0 0, 0, 2 4 2, 4 3 9 3, 9
(Section.3: Basic Graphs and Symmetry).3.4 TIP : The graph never falls below the x-axis, because squares of real numbers are never negative. Look at the table. Each pair of opposite x values yields a common function value f x ( ), or y. ( ) on the graph has a ( ) that is also on the graph. These Graphically, this means that every point x, y mirror image partner x, y mirror image pairs are symmetric about the y-axis. We say that f is an even function. (Why?) A function f is even f ( x)= f ( x), x Dom( f ) The graph of y = f ( x) is symmetric about the y-axis. Example (Even Function: Proof) Solution Let f ( x)= x 2. Prove that f is an even function. Dom( f )=. x, f ( x)= ( x) 2 = x 2 ( ) = f x Q.E.D. (Latin: Quod Erat Demonstrandum) This signifies the end of a proof. It means that which was to have been proven, shown, or demonstrated. TIP 2: Think: If we replace x with x ( ) as the input, we obtain equivalent outputs.
(Section.3: Basic Graphs and Symmetry).3.5 PART F: POWER FUNCTIONS WITH POSITIVE, EVEN POWERS and INTERSECTION POINTS The term even function comes from the following fact: If f ( x)= x n, where n is an even integer, then f is an even function. The graph of y = x 2 is called a parabola (see Chapters 2 and 0). The graphs of y = x 4, y = x 6, etc. resemble that parabola, although they are not called parabolas. We will discuss the cases with nonpositive exponents later. How do these graphs compare? For example, let f ( x)= x 2 and g( x)= x 4. Compare the graphs of f and g. Their relationship when x > is unsurprising: x f ( x) x 2 g( x) x 4 2 4 6 3 9 8 4 6 256 As expected, x 4 > x 2 if x >. As a result, the graph of y = x 4 lies above the graph of y = x 2 on the x-interval (, ).
(Section.3: Basic Graphs and Symmetry).3.6 However, their relationship on the x-interval 0, might be surprising: x f ( x) x 2 g( x) x 4 0 0 0 0. 0.0 0.000 3 2 9 4 8 6 WARNING : As it turns out, x 4 < x 2 on the x-interval ( 0,). As a result, the graph of y = x 4 lies below the graph of y = x 2 on that x-interval. Since the graphs have the points 0, 0 points are intersection points. ( ) and (, ) in common, those Graphically, here s what we have (so far) on the x-interval 0, ). Below, f ( x)= x 2 and g( x)= x 4.
(Section.3: Basic Graphs and Symmetry).3.7 How can we quickly get the other half of the picture? Exploit symmetry! f and g are both even functions, so their graphs are symmetric about the y-axis. Observe that (, ) is our third intersection point. In calculus, you might find the area of one or both of those tiny regions bounded (trapped) by the graphs. Let h( x)= x 6. How does the graph of h below compare? The graph of h rises even faster than the others as we move far away from x = 0, but it is even flatter than the others close to x = 0.
(Section.3: Basic Graphs and Symmetry).3.8 PART G: POWER FUNCTIONS WITH POSITIVE, ODD POWERS and ODD FUNCTIONS Let f ( x)= x 3. We will construct a table and graph f. x ( ) Point x f ( x) Point ( ) 0 0 ( 0, 0) ( ) (, ) ( ) 2 8 ( 2, 8) ( ) 3 27 ( 3, 27) f x 0 0 0, 0, 2 8 2, 8 3 27 3, 27
(Section.3: Basic Graphs and Symmetry).3.9 Look at the table. Each pair of opposite x values yields opposite function values. That is, f x ( ) and f ( x) are always opposites. ( ) on the graph has a ( ) on the other side of the origin. Graphically, this means that every point x, y mirror image partner x, y The two points are separated by a 80 rotation (a half revolution) about the origin. These mirror image pairs are symmetric about the origin. We say that f is an odd function. (Why?) A function f is odd f ( x)= f ( x), x Dom( f ) The graph of y = f ( x) is symmetric about the origin. In other words, if the graph of f is rotated 80 about the origin, we obtain the same graph. Example 2 (Odd Function: Proof) Let f ( x)= x 3. Prove that f is an odd function. Solution Dom( f )=. x, Q.E.D. f ( x)= ( x) 3 = x 3 ( ) = x 3 ( ) = f x TIP 3: Think: If we replace x with x ( ) as the input, we obtain opposite outputs.
The term odd function comes from the following fact: (Section.3: Basic Graphs and Symmetry).3.0 If f ( x)= x n, where n is an odd integer, then f is an odd function. The graphs of y = x 5, y = x 7, etc. resemble the graph of y = x 3. In Part C, we saw that the graph of y = x is a line. We will discuss the cases with negative exponents later. How do these graphs compare? For example, let f ( x)= x 3 and g( x)= x 5. Compare the graphs of f and g. Based on our experience from Part F, we expect that the graph of g rises or falls even faster than the graph of f as we move far away from x = 0, but it is even flatter than the graph of f close to x = 0. WARNING 2: Zero functions are functions that only output 0 (Think: f ( x)= 0). Zero functions on domains that are symmetric about 0 on the real number line are the only functions that are both even and odd. (Can you show this?) WARNING 3: Many functions are neither even nor odd.
(Section.3: Basic Graphs and Symmetry).3. PART H : f ( x)= x 0 Let f ( x)= x 0. What is f ( 0)? It is agreed that 0 2 = 0 and 2 0 =, but what is 0 0? Different sources handle the expression 0 0 differently. If 0 0 is undefined, then f ( x)= ( x 0), and f has the graph below. There is a hole at the point ( 0,). There are many reasons to define 0 0 to be. For example, when analyzing polynomials, it is convenient to have x 0 = for all real x without having to consider x = 0 as a special case. Also, this will be assumed when we discuss the Binomial Theorem in Section 9.6. Then, f ( x)= on, and f has the graph below. In calculus, 0 0 is an indeterminate limit form. An expression consisting of a base approaching 0 raised to an exponent approaching 0 may, itself, approach a real number (not necessarily 0 or ) or not. The expression 0 0 is called indeterminate by some sources. In any case, f is an even function.
(Section.3: Basic Graphs and Symmetry).3.2 PART I: RECIPROCAL FUNCTION and POWER FUNCTIONS WITH NEGATIVE, ODD POWERS Let f ( x)= x ( or x ). We call f a reciprocal function, because its output is the reciprocal (or multiplicative inverse) of the input. We will carefully construct the graph of f. Let s construct a table for x. x 0 00 f ( x), or x 0 00 0 + The 0 + notation indicates an approach to 0 from greater numbers, without reaching 0. The table suggests the following graph for x : The x-axis is a horizontal asymptote ( HA ) of the graph. An asymptote is a line that a curve approaches in a long-run or explosive sense. The distance between them approaches 0. Asymptotes are often graphed as dashed lines, although some sources avoid dashing the x- and y-axes. Horizontal and vertical asymptotes will be formally defined in Section 2.9.
Let s now construct a table for 0 < x. (Section.3: Basic Graphs and Symmetry).3.3 x 0 + 00 0 f ( x), or x 00 0 We write: x as x 0+ ( approaches infinity as x x approaches 0 from the right, or from greater numbers ). In the previous table, x 0+ as x. Graphically, x approaches 0 from above, though we say from the right. We will revisit this notation and terminology when we discuss limits in calculus in Section.5. We now have the following graph for x > 0 : The y-axis is a vertical asymptote ( VA ) of the graph. How can we quickly get the other half of the picture? Exploit symmetry! f is an odd function, so its graph is symmetric about the origin. TIP 4: Reciprocals of negative real numbers are negative real numbers. 0 has no real reciprocal.
(Section.3: Basic Graphs and Symmetry).3.4 The graph exhibits opposing behaviors about the vertical asymptote ( VA ). The function values increase without bound from the right of the VA, and they decrease without bound from the left of the VA. The graph of y = x ( or x ), or xy =, is called a hyperbola (see Chapter 0). The graphs of y = ( or x 3 ), y = ( or x 5 ), etc. resemble that hyperbola, but x 3 x 5 they are not called hyperbolas. Below, f ( x)= x yields the blue graph; g( x)= yields the red graph. 3 x The graph of g approaches the x-axis more rapidly as x and as x. The graph of g approaches the y-axis more slowly as x 0 + and as x 0 ( as x approaches 0 from the left, or from lesser numbers ). This is actually because the values of g explode more rapidly. When we investigate the graph of y = in Part J, we will understand these 2 x behaviors better.
(Section.3: Basic Graphs and Symmetry).3.5 PART J: POWER FUNCTIONS WITH NEGATIVE, EVEN POWERS Let h( x)= ( or x 2 ). We will compare the graph of h to the graph of y = x 2 x. Let s construct a table for x. x 0 00 f ( x), or x 0 00 0 h( x), or x 2 00 0,000 0 This suggests that the graph of h approaches the x-axis more rapidly as x. The table suggests the following graphs for x : The x-axis is a horizontal asymptote ( HA ) of the graph of h. Let s now construct a table for 0 < x. x 0 + 00 0 f ( x), or x 00 0 h( x), or 0,000 00 x 2 This suggests that the graph of h approaches the y-axis more slowly as x 0 +. This is actually because the values of h explode more rapidly.
We now have the following graphs for x > 0 : (Section.3: Basic Graphs and Symmetry).3.6 The y-axis is a vertical asymptote ( VA ) of the graph of h. How can we quickly get the other half of the graph of h? Exploit symmetry! h is an even function, so its graph is symmetric about the y-axis. TIP 5: This graph lies entirely above the x-axis, because is always 2 x positive in value for nonzero values of x. The graph exhibits symmetric behaviors about the vertical asymptote ( VA ). The function values increase without bound from the left and from the right of the VA. The graphs of y = x 4 or x 4, y = x 6 or x 6, etc. resemble the graph above.
PART K: SQUARE ROOT FUNCTION (Section.3: Basic Graphs and Symmetry).3.7 Let f ( x)= x ( or x /2 ). We discussed the graph of f in Section.2. WARNING 4: f is not an even function, because it is undefined for x < 0. 4 The graphs of y = x ( or x /4 6 ), y = x ( or x /6 ), etc. resemble this graph, as do the graphs of y = x ( 4 3 or x 3/4 ), y = x ( 8 5 or x 5/8 ), etc. (See Footnote.) PART L: CUBE ROOT FUNCTION 3 Let f ( x)= x ( or x /3 ). The graph of f resembles the graph in Part K for x 0. WARNING 5: The cube root of a negative real number is a negative real number. Dom f ( )=. f is an odd function; its graph is symmetric about the origin. 5 The graphs of y = x ( or x /5 7 ), y = x ( or x /7 ), etc. resemble this graph, as do the graphs of y = x ( 5 3 or x 3/5 ), y = x ( 9 5 or x 5/9 ), etc. (See Footnote 2.)
(Section.3: Basic Graphs and Symmetry).3.8 PART M : f ( x)= x 2/3 Let f ( x)= 3 x 2 ( or x 2/3 ). The graph of f resembles the graphs in Parts K and L for x 0. f is an even function; its graph below is symmetric about the y-axis. WARNING 6: Some graphing utilities omit the part of the graph to the left of the y-axis. In calculus, we will call the point at ( 0, 0) a cusp, because: it is a sharp turning point for the graph, and as we approach the point from either side, we approach ± ( ) infinite steepness. The graphs of y = x ( 5 2 or x 2/5 ), y = x ( 7 4 or x 4/7 ), etc. resemble the graph above. (See Footnote 3.)
PART N: ABSOLUTE VALUE FUNCTION We discussed the absolute value operation in Section 0.4. (Section.3: Basic Graphs and Symmetry).3.9 The piecewise definition of the absolute value function (on ) is given by: f ( x)= x = x, if x 0 x, if x < 0 We will discuss more piecewise-defined functions in Section.5. f is an algebraic function, because we can write: f ( x)= x = x 2. WARNING 7: Writing x 2 as x 2/2 would be inappropriate if it is construed as x, which would not be equivalent for x < 0, or as ( x ) 2, which has domain 0, ). (See Footnote 4.) f is an even function, so its graph will be symmetric about the y-axis. The graph of y = x for x 0 has a mirror image in the graph of y = x for x 0. In calculus, we will call the point at ( 0, 0) a corner, because: the graph makes a sharp turn there, and the point is not a cusp. (A corner may or may not be a turning point where the graph changes from rising to falling, or vice-versa.)
(Section.3: Basic Graphs and Symmetry).3.20 PART O: UPPER SEMICIRCLES In Section.2, we saw that the graph of x 2 + y 2 = 9 y 0 circle of radius 3 centered at ( 0, 0). Solving for y, we obtain: y = 9 x 2. ( ) is the upper half of the More generally, the graph of x 2 + y 2 = a 2 ( y 0), where a > 0, is an upper semicircle of radius a. Solving for y, we obtain: y = a 2 x 2. Let f x ( )= a 2 x 2. f is an even function, so its upper semicircular graph below is symmetric about the y-axis.
PART P: A GALLERY OF GRAPHS (Section.3: Basic Graphs and Symmetry).3.2 TIP 6: If you know the graphs well, you don t have to memorize the domains, ranges, and symmetries. They can be inferred from the graphs. In the Domain and Range column, \{} 0 denotes the set of nonzero real numbers. In interval form, \{} 0 is (,0) ( 0, ). Function Rule Type of Function (Sample) Graph Domain; Range Even/Odd; Symmetry f ( x)= c Constant ; c {} Even; y-axis f ( x)= x Identity (Type of Linear) ; Odd; origin f ( x)= mx + b ( m 0) Linear ; Odd b = 0; then, origin f ( x)= x 2 ( x n : n 2, even) Power ; 0, ) Even; y-axis f ( x)= x 3 ( x n : n 3, odd) Power ; Odd; origin f ( x)= x 0 Power See Part H See Part H Even; y-axis f ( x)= x or x x n : n < 0, odd ( ) Power \{}; 0 {} \ 0 Odd; origin f ( x)= x 2 or x 2 ( x n : n < 0, even) Power \{}; 0 ( 0, ) Even; y-axis
(Section.3: Basic Graphs and Symmetry).3.22 Function Rule Type of Function (Sample) Graph Domain; Range Even/Odd; Symmetry f ( x)= x /2 or x Power n ( x : n 2, even) 0, ); 0, ) Neither f ( x)= x /3 3 or x Power n ( x : n 3, odd) ; Odd; origin f ( x)= x 2/3 Power ; 0, ) Even; y-axis f ( x)= x Absolute Value (Algebraic) ; 0, ) Even; y-axis f ( x)= a 2 x 2 ( a > 0) (Type of Algebraic) a, a ; 0, a Even; y-axis
(Section.3: Basic Graphs and Symmetry).3.23 FOOTNOTES. Power functions with rational powers of the form odd even. Let f ( x)= x N / D, where N is an odd and positive integer, and D is an even and positive integer. f ( x)= x /2 f ( x)= x 3/2 If N D is a proper fraction (where N < D ), then the graph of f is concave down and resembles the graph on the left. Examples: f ( x)= x ( or x /2 ), f ( x)= 4 x ( 3 or x 3/4 ). If N D is an improper fraction (where N > D ), then the graph of f is concave up and resembles the graph on the right. Examples: f ( x)= x ( 3 or x 3/2 ), f ( x)= 4 x ( 7 or x 7/4 ). 2. Power functions with rational powers of the form odd odd. Let f ( x)= x N / D, where N and D are both odd and positive integers. f ( x)= x /3 f ( x)= x 3/3 = x f ( x)= x 9/3 = x 3 If N D is a proper fraction, then the graph of f resembles the leftmost graph. 3 Examples: f ( x)= x ( or x /3 ), f ( x)= 5 x ( 3 or x 3/5 ). If N D is an improper fraction where N > D, then the graph of f resembles the rightmost graph. For example, f ( x)= x 9/3 = x 3. If N = D ( N D is still improper), then we obtain the line y = x (see the middle graph) as a borderline case. For example, f ( x)= x 3/3 = x.
(Section.3: Basic Graphs and Symmetry).3.24 3. Power functions with rational powers of the form even odd. Let f x ( )= x N / D, where N is an even and positive integer, and D is an odd and positive integer. f ( x)= x 2/3 f ( x)= x 6/3 = x 2 If N D is a proper fraction, then the graph of f resembles the graph on the left. Examples: f ( x)= 3 x ( 2 or x 2/3 ), f ( x)= 7 x ( 4 or x 4/7 ). If N D is an improper fraction, then the graph of f resembles the graph on the right, where f ( x)= 3 x 6 = x 6/3 = x 2. 4. Power functions with rational powers of the form even even. Let f ( x)= x N / D, where N and D are both even and positive integers. Different interpretations of x N / D lead to different approaches to Dom( f ). For example, let f ( x)= x 2/6. If x 0, then f ( x)= x 2/6 = x /3 3, or x. If x 2/6 is interpreted as 6 x 2, then x 2/6 is real-valued, even if x < 0. Under this interpretation, Dom( f )=. If x 2/6 6 is interpreted as x Under this interpretation, Dom( f )= 0, ). ( ) 2, then x 2/6 is not real-valued when x < 0.
(Section.4: Transformations).4. SECTION.4: TRANSFORMATIONS LEARNING OBJECTIVES Know how to graph transformations of functions. Know how to find an equation for a transformed basic graph. Use graphs to determine domains and ranges of transformed functions. PART A: DISCUSSION Variations of the basic functions from Section.3 correspond to variations of the basic graphs. These variations are called transformations. Graphical transformations include rigid transformations such as translations ( shifts ), reflections, and rotations, and nonrigid transformations such as vertical and horizontal stretching and squeezing. Sequences of transformations correspond to compositions of functions, which we will discuss in Section.6. After this section, we will be able to graph a vast repertoire of functions, and we will be able to find equations for many transformations of basic graphs. We will relate these ideas to the standard form of the equation of a circle with center ( h, k), which we saw in Section 0.3. In the Exercises, the reader can revisit the Slope-Intercept Form of the equation of a line, which we saw in Section 0.4. We will use these ideas to graph parabolas in Section 2.2 and conic sections in general in Chapter 0, as well as trigonometric graphs in Chapter 4. Thus far, y and f x ( ) have typically been interchangeable. This will no longer be the case in many of our examples.
(Section.4: Transformations).4.2 PART B: TRANSLATIONS ( SHIFTS ) Translations ( shifts ) are transformations that move a graph without changing its shape or orientation. Let G be the graph of y = f ( x). Let c be a positive real number. Vertical Translations ( Shifts ) The graph of y = f ( x)+c is G shifted up by c units. We are increasing the y-coordinates. The graph of y = f ( x) c is G shifted down by c units. Horizontal Translations ( Shifts ) The graph of y = f x c The graph of y = f x+c Example (Translations) ( ) is G shifted right by c units. ( ) is G shifted left by c units. Let f ( x)= x. Its graph, G, is the center graph in purple below.
A table can help explain how these translations work. In the table, und. means undefined. x f ( x) x f ( x)+ 2 x + 2 f ( x) 2 x 2 (Section.4: Transformations).4.3 f ( x 2) x 2 f ( x+ 2) x + 2 3 und. und. und. und. und. 2 und. und. und. und. 0 und. und. und. und. 0 0 2 2 und. 2 3 und. 3 2 2 2 + 2 2 2 0 2 3 3 3 + 2 3 2 5 How points change y-coords. increase 2 units y-coords. decrease 2 units x-coords. increase 2 units x-coords. decrease 2 units G moves UP DOWN RIGHT LEFT Example 2 (Finding Domain and Range; Revisiting Example ) We can infer domains and ranges of the transformed functions in Example from the graphs and the table in Example. Let f ( x)= x. Then, Dom( f )= Range( f )= 0, ). Let g( x)= x + 2 x 2 x 2 x + 2 ( ) Dom g Think: x Range g ( ) Think: y 0, ) 0, ) 2, ) 2, ) 2, ) 2, ) 0, ) 0, )
WARNING : Many people confuse the horizontal shifts. (Section.4: Transformations).4.4 Compare the x-intercepts of the graphs of y = x and y = x 2. The x-intercept is at x = 0 for the first graph, while it is at x = 2 for the second graph. The fact that the point 0, 0 that the point 2, 0 ( ) lies on the first graph implies ( ) lies on the second graph. More generally: The point ( a, b) lies on the first graph the point ( a + 2, b) lies on the second graph. Therefore, the second graph is obtained by shifting the first graph to the right by 2 units. PART C: REFLECTIONS Reflections Let G be the graph of y = f ( x). The graph of y = f x The graph of y = f x The graph of y = f x Example 3 (Reflections) ( ) is G reflected about the x-axis. ( ) is G reflected about the y-axis. ( ) is G reflected about the origin. This corresponds to a 80 rotation (half revolution) about the origin. It combines both transformations above, in either order. Again, let f ( x)= x.
A table can help explain how these reflections work. In the table, und. means undefined. x f ( x) x f ( x) x (Section.4: Transformations).4.5 f ( x) x f ( x) 3 und. und. 3 3 2 und. und. 2 2 und. und. 0 0 0 0 0 und. und. 2 2 2 und. und. 3 3 3 und. und. x Points are reflected about x-axis y-axis Both, or origin Example 4 (Finding Domain and Range; Revisiting Example 3) We can infer domains and ranges of the transformed functions in Example 3 from the graphs and the table in Example 3. Let f ( x)= x. Then, Dom( f )= Range( f )= 0, ). Let g( x)= x x x ( ) Dom g Think: x Range g ( ) Think: y 0, ) (,0 (,0 (,0 0, ) (,0 WARNING 2: x is defined as a real value for nonpositive real values of x, because the opposite of a nonpositive real value is a nonnegative real value.
(Section.4: Transformations).4.6 Example 5 (Reflections and Symmetry) Let f ( x)= x 2. The graph of f is below. The graph is its own reflection about the y-axis, because f is an even function. The graphs of y = f x ( ) and y = f ( x) are the same: f ( x)= ( x) 2 = x 2. Thus, the graph is symmetric about the y-axis. Example 6 (Reflections and Symmetry) Let f ( x)= x 3. The graph of f is below. The graph is its own reflection about the origin, because f is an odd function. The graphs of y = f x ( ) and y = f ( x) are the same: f ( x)= x ( ) 3 = x 3. The graph is symmetric about the origin.
(Section.4: Transformations).4.7 PART D: NONRIGID TRANSFORMATIONS; STRETCHING AND SQUEEZING Nonrigid transformations can change the shape of a graph beyond a mere reorientation, perhaps by stretching or squeezing, unlike rigid transformations such as translations, reflections, and rotations. If f is a function, and c is a real number, then cf is called a constant multiple of f. The graph of y = cf ( x) is: a vertically stretched version of G if c > a vertically squeezed version of G if 0 < c < The graph of y = f ( cx) is: a horizontally squeezed version of G if c > a horizontally stretched version of G if 0 < c < If c < 0, then perform the corresponding reflection either before or after the vertical or horizontal stretching or squeezing. WARNING 3: Just as for horizontal translations ( shifts ), the cases involving horizontal stretching and squeezing may be confusing. Think of c as an aging factor. Example 7 (Vertical Stretching and Squeezing) Let f ( x)= x. First consider the form y = cf ( x).
(Section.4: Transformations).4.8 For any x-value in 0, ), such as, the corresponding y-coordinate for the y = x graph is doubled to obtain the y-coordinate for the y = 2 x graph. This is why there is vertical stretching. Similarly, the graph of y = x exhibits vertical squeezing, because the 2 y-coordinates have been halved. Example 8 (Horizontal Stretching and Squeezing; Revisiting Example 7) Again, let f ( x)= x. Now consider the form y = f ( cx). ( ) is the graph of y = 2 x in blue, because: The graph of y = f 4x f ( 4x)= 4x = 2 x. The vertical stretching we described in Example 7 may now be interpreted as a horizontal squeezing. (This is not true of all functions.) The function value we got at x = we now get at x = 4. The graph of y = f 4 x is the graph of y = 2 x in red, because: f 4 x = 4 x = x. The vertical squeezing we described in Example 7 2 may now be interpreted as a horizontal stretching. The function value we got at x = we now get at x = 4.
(Section.4: Transformations).4.9 PART E: SEQUENCES OF TRANSFORMATIONS Example 9 (Graphing a Transformed Function) Graph y = 2 x + 3. Solution We may want to rewrite the equation as y = indicate the vertical shift. x + 3 + 2 to more clearly We will build up the right-hand side step-by-step. Along the way, we transform the corresponding function and its graph. We begin with a basic function with a known graph. (Point-plotting should be a last resort.) Here, it is a square root function. Let f ( x)= x. Basic Graph: y = x Graph: y = x + 3 Begin with: f ( x)= x Transformation: f 2 ( x)= f ( x + 3) Effect: Shifts graph left by 3 units Graph: y = x + 3 Graph: y = x + 3 + 2 Transformation: f 3 ( x)= f 2 ( x) Transformation: f 4 ( x)= f 3 ( x)+ 2 Effect: Reflects graph about x-axis Effect: Shifts graph up by 2 units
(Section.4: Transformations).4.0 WARNING 4: We are expected to carefully trace the movements of any key points on the developing graphs. Here, we want to at least trace the movements of the endpoint. We may want to identify intercepts, as well. ( )? Why is the y-intercept of our final graph at 2 3, or at 0, 2 3 Why is the x-intercept at, or at (, 0)? (Left as exercises for the reader.) Example 0 (Finding an Equation for a Transformed Graph) Find an equation for the transformed basic graph below. Solution The graph appears to be a transformation of the graph of the absolute value function from Section.3, Part N. Basic graph: y = x Begin with: f ( x)= x
(Section.4: Transformations).4. There are different strategies that can lead to a correct equation. Strategy (Raise, then reflect) Effect: Shifts graph up by unit Effect: Reflects graph about x-axis Transformation: f 2 ( x)= f ( x)+ Transformation: f ( x)= f 2 ( x) ( ) Graph: y = x + Graph: y = x + WARNING 5: It may help to write f ( x)= f 2 ( x), since it reminds us to insert grouping symbols. Possible answers: f ( x)= ( x + ), or f ( x)= x. Strategy 2 (Reflect, then drop) Effect: Reflects graph about x-axis Effect: Shifts graph down by unit Transformation: f 2 ( x)= f ( x) Transformation: f ( x)= f 2 ( x) Graph: y = x Graph: y = x Possible answer: f ( x)= x, which we saw in Strategy.
(Section.4: Transformations).4.2 Strategy 3 (Switches the order in Strategy 2, but this fails!) Basic graph: y = x Begin with: f ( x)= x Effect: Shifts graph down by unit Effect: Reflects graph about x-axis Transformation: f 2 ( x)= f ( x) Transformation: f 3 ( x)= f 2 ( x) ( ) Graph: y = x Graph: y = x Observe that y = ( x ) is not equivalent to our previous answers. WARNING 6: The order in which transformations are applied can matter, particularly when we mix different types of transformations.
(Section.4: Transformations).4.3 PART F: TRANSLATIONS THROUGH COORDINATE SHIFTS Translations through Coordinate Shifts A graph G in the xy-plane is shifted h units horizontally and k units vertically. If h < 0, then G is shifted left by h units. If k < 0, then G is shifted down by k units. To obtain an equation for the new graph, take an equation for G and: Replace all occurrences of x with ( x h), and Replace all occurrences of y with ( y k). Example (Translating a Circle through Coordinate Shifts; Revisiting Section 0.3) Solution ( ) We want to translate the circle in the xy-plane with radius 3 and center 0, 0 so that its new center is at ( 2,). Find the standard form of the equation of the new circle. We take the equation x 2 + y 2 = 9 for the old black circle and: Replace x with ( x ( 2) ),or( x + 2), and Replace y with ( y ). This is because we need to shift the black circle left 2 units and up unit to obtain the new red circle.
(Section.4: Transformations).4.4 Answer: ( x + 2) 2 + ( y ) 2 = 9. We will use this technique in Section 2.2 and Chapter 0 on conic sections. Equivalence of Translation Methods for Functions Consider the graph of y = f ( x). A coordinate shift of h units horizontally and k units vertically yields an equation that is equivalent to one we would have obtained from our previous approach: ( ) y k = f x h y = f ( x h)+ k Think: h, k, if h and k are positive numbers. We will revisit this form when we study parabolas in Section 2.2. Example 2 (Equivalence of Translation Methods for Functions) We will shift the first graph to the right by 2 units and up unit. Graph of y = x 2 Graph of y = x 2 ( ) 2, or ( ) 2 + y = x 2