Example 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs

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1 Quadratic Functions: - functions defined by quadratic epressions (a 2 + b + c) o the degree of a quadratic function is ALWAYS 2 - the most common way to write a quadratic function (and the way we have seen quadratics in the past) is polynomial form o f() = a 2 + b + c - the graph of a quadratic function is a parabola ( or ) o in order to be the graph of a function, the parabola must be vertical a parabola opening sideways would not pass the vertical line test and as a result would not be a function o the leading coefficient a determines whether the parabola opens up or down if a > 0 the parabola opens up ( ) if a < 0 the parabola opens down ( ) Verte: - the turning point of a parabola Eample 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: f() f() = V ( 1 2, 25 4 )

2 a. Find the domain (the set of inputs or -values) and range (the set of outputs or function values) of the function f() = f() Domain: (, ) Range: [ 25 4, ) V ( 1 2, 25 4 ) The domain can found algebraically or graphically. Algebraically we can see that the quadratic function f() = has no square roots and/or fractions, so there is nothing restricting its domain. Graphically we can see the that the parabola continues to move to the left ( ) and to the right ( ), so the domain goes on forever in both directions. The range can be found graphically as well; we can see that the smallest output of the function is the ycoordinate of the verte ( 25 ), and from the graph just 4 continues to rise to infinity.

3 b. Find the zeros of the function f() = (f() = 0 when =? ) f() V ( 1 2, 25 4 ) f() = 0 when = 3, 2 To find the zeros of a function algebraically, simply replace f() with 0 and solve the quadratic equation (in this case the quadratic equation would be 0 = 2 + 6). For a review of how to solve quadratic equations, take a look at Lessons 12, 13, and 14. In this situation, we could also find the zeros graphically by identifying the -values where the graph crosses the -ais. The graph of the quadratic function f() = crosses the -ais at = 3 and at = 2. We will not always be able to identify zeros graphically, but sometimes it is possible.

4 c. List the positive and negative intervals of f() = What values make f() > 0? What values make f() < 0? f() V ( 1 2, 25 4 ) The inputs ( values) that make f() > 0 are (, 3) (2, ) The inputs ( values) that make f() < 0 are ( 3, 2) Hint: Consider using a piece of paper to cover up the top of the parabola (everything above the -ais) or bottom of the parabola (everything below the -ais) in order to help identify the negative or positive intervals of the function. Also, keep in mind that even though we are finding the intervals where the outputs are positive or negative, we epress those intervals in terms of the inputs. Be aware that an answer such as ( 3, 2) could be an interval, as it was here, and it could also be an ordered, such a quadratic function with a verte at the point ( 3, 2). This ambiguity sometimes causes confusion when working with quadratic functions.

5 d. List the increasing and decreasing intervals of f() = What values make f() rise? What values make f() fall? f() V ( 1 2, 25 4 ) Increasing intervals represent the inputs that make the graph rise, or the intervals where the function has a positive slope. Decreasing intervals represent the inputs that make the graph fall, or the intervals where the function has a negative slope. Also, consider using a piece of paper to cover up the right half (everything to the left of the verte) or left half (everything to the right of the verte) of the parabola in order to help identify the decreasing or increasing intervals of the function. The inputs ( values) that make f() rise are ( 1 2, ) The inputs ( values) that make f() fall are (, 1 2 )

6 e. List the intercepts of the function f() = as order pairs (remember that intercepts are points). f() V ( 1 2, 25 4 ) intercepts: ( 3, 0), (2, 0) y intercepts: (0, 6) -intercepts are the points (or the point) where a graph touches or crosses the -ais. Remember that every point on the -ais has a ycoordinate of 0. y-intercepts are the points (or the point) where a graph touches or crosses the y-ais. Remember that every point on the y-ais has an -coordinate of 0. Be sure to always epress intercepts as ordered pairs.

7 What the Verte Tells Us About a Quadratic Function: - the -coordinate of the verte determines the where the graph changes from increasing to decreasing or decreasing to increasing o if the parabola opens up, the graph is decreasing from (, ) and increasing from (, ) o if the parabola opens down, the graph is increasing from (, ) and decreasing from (, ) - the -coordinate of the verte also determines the line of symmetry o the line of symmetry is the vertical line that cuts the parabola in half o the equation of the line of symmetry is = -coordinate - the y-coordinate of the verte is either the maimum function value or minimum function value (largest or smallest output) o if a parabola opens up, the verte is a minimum and the ycoordinate is smallest function value o if a parabola opens down, the verte is a maimum and the y-coordinate is largest function value - the y-coordinate of the verte also determines the range of a quadratic function Verte: ( 1, 2) This function is decreasing on the interval (, 1) and increasing on the interval ( 1, ) because the graph changes direction at = 1. Verte: (4, 2) This function is has a range of (, 2], because its verte is at the top of the parabola (so 2 is the largest output) and it is opening down (going to negative infinity). o if the verte is the minimum function value, the range is the y-coordinate to infinity [y, ) because the parabola opens up o if the verte is the maimum function value, the range is negative infinity to the y-coordinate (, y] because the parabola opens down

8 If the graph of a quadratic function does not touch or cross the -ais, then the function has no zeros and no -intercepts. Eamples of this are given below. Even without the graphs provided, we could still determine that the functions are never equal to zero by setting each epression equal to zero and solving. In each case we will end up with imaginary solutions, which would indicate that there are no real zeros. Notice that these quadratic functions are either always positive or always negative. If the graph of the function does not cross the -ais, the sign of the function (positive or negative) does not change. f() = f() a. g() = 2 = 1 2, g() = f() V(1, 2) V( 2, 1) Notice that the graph of the function f never touches or crosses the -ais. Therefore there are no zeros, no -intercepts, and no negative intervals, because f() > 0 for every input. The graph of the function g never touches or crosses the -ais because g() < 0 for every input. Zeros: NONE intercepts: NONE g() < 0: NONE

9 Eample 2: Given below is the graph of the quadratic function g. Use the function and its graph to find the following: g() = f() V(2, 10) a. Find the domain (the set of inputs or -values) and range (the set of outputs or function values) of the function g() = b. Find the zeros of the function (g() = 0 when =? ). c. List the positive and negative intervals of. What values make g() > 0? What values make g() < 0? d. List the increasing and decreasing intervals of. What values make g() rise? What values make g() fall? e. List the intercepts of the function as order pairs (remember that intercepts are points).

10 Eample 3: Given below is the graph of the quadratic function h. Use the function and its graph to find the following: h() = f() V(2,0) a. Find the domain (the set of inputs or -values) and range (the set of outputs or function values) of the function h() = b. Find the zeros of the function (h() = 0 when =? ). c. List the positive and negative intervals of. What values make h() > 0? What values make h() < 0? d. List the increasing and decreasing intervals of. What values make h() rise? What values make h() fall? e. List the intercepts of the function as order pairs (remember that intercepts are points).

11 Eample 4: Given below is the graph of the quadratic function j. Use the function and its graph to find the following: j() = 2 2 f() V(0, 2) a. Find the domain (the set of inputs or -values) and range (the set of outputs or function values) of the function j() = b. Find the zeros of the function (j() = 0 when =? ). c. List the positive and negative intervals of. What values make j() > 0? What values make j() < 0? d. List the increasing and decreasing intervals of. What values make j() rise? What values make j() fall? e. List the intercepts of the function as order pairs (remember that intercepts are points).

12 Answers to Eamples: 1a. D: (, ), R: [ 25 4, ) ; 1b. f() = 0 when = 3, 2 ; 1c. f() > 0: (, 3) (2, ), f() < 0: ( 3, 2), ; 1g. : ( 1 2, ), : (, 1 2 ) ; 1e. ints: ( 3, 0), (2, 0), y int: (0, 6) ; 2a. D: (, ), R: (, 10] ; 2b. g() = 0 when = 2 ± 10 ; 2c. g() > 0: (2 10, ), g() < 0: (, 2 10) (2 + 10, ) ; 2d. : (, 2), : (2, ) ; 2e. ints: (2 10, 0), (2 + 10, 0), y int: (0, 6) ; 3a. D: (, ), R: (, 0] ; 3b. h() = 0 when = 2 ; 3c. h() > 0: NONE, h() < 0: (, 2) (2, ); 3d. : (, 2), : (2, ) ; 3e. int: (2, 0), y int: (0, 4) ; 4a. D: (, ), R: (, 2] ; 4b. j() 0 ; 4c. j() > 0: NONE, j() < 0: (, ); 4d. : (, 0), : (0, ) ; 4e. ints: NONE, y int: (0, 2) ;