NOTES: ALGEBRA FUNCTION NOTATION
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1 STARTER: 1. Graph f by completing the table. f, y NOTES: ALGEBRA 4.1 FUNCTION NOTATION y. Graph f 4 4 f 4 4, y y A Brief Review of Function Notation We will be using function notation more often. The equation y = +7 can also be written as f = + 7 FUNCTION NOTATION y = f We say This means that y depends on. Sometimes, we refer to f as In this case, the parenthesis do not mean multiplication. The represents the number we are plugging in to the function. f represents what we will be getting out. f means We don t have to name our functions f, any letter will do!
2 Eample: This is a graph of a function. Think of it as a map. If you have to find f6, 6 is what we are plugging in. So, we look for 6 on the -ais. We then find where our graph is at 6. f6 = f1= f0 = f0 = Eample: Use the graph below to answer the following questions: a. f = b. f0 = c. f4 = d. f9 = e. If f = 1, what is? f. If f = 4, what is? There are answers...
3 Eample: Consider the function g = Find g1, g, and g0.. For what value of does g = 7? Eample: Given f. 1. Find the values of f 0, f 7, f 5.. For what value of does f 10? Eample: Evaluate the following given f = and g = f 1 =. f 0 =. g4 = 4. g1 =
4 Eample: The graph of the function h is drawn below. Use it to find the following: 1... h = h = h 1 = h = 1 = 4. If, h = 0 = = 5. If, OR Eample: Use the graph of f below to evaluate the following. 1. f 4. f. f 0 4. f 1, Eample: Isaac lives miles away from his school. School ended at pm and Isaac began his walk home with his friend Tate who lives 1 mile from the school in the direction of Isaac s house. Isaac stayed at Tate s house for a while and then started home. On the way, he stopped at the library then he hurried home. The graph at the below is the piece-wise defined function that shows Isaac s distance from home during the time it took him to arrive home. 1. How much time passed between school ending and Isaac s arrival home?. How long did Isaac stay at Tate s house?. How far is the library from Isaac s house? 4. Where was Isaac hours after school ended? 5. Rewrite #4 in function notation.
5 NOTES: ALGEBRA 4. PROPERTIES OF QUADRATICS STARTER: 1. Graph the line f 4. Find the domain and range of the following function. y y. Given f 1 5, find f 4 4. Solve for. 6y 4 INTRODUCTION TO QUADRATIC FUNCTIONS Recall: The Quadratic Parent Function Equation: Graph: Description of Graph: General Equation: Verte Form: Intercept Form: Domain: Range: DETERMINE WHETHER THE GIVEN FUNCTION IS LINEAR, QUADRATIC, OR NEITHER. 1. f g 1. f
6 KEY FEATURES OF A QUADRATIC The -intercept also called a zero is the point where a graph crosses or touches the -ais. o It is the ordered pair, where is a real number. o The -intercept is where. The y-intercept is the point where a graph crosses or touches they y-ais. o It is the ordered pair, where y is a real number. o The y-intercept is where. -intercept y-intercept The verte of a parabola is the high point or low point in the curve. The verte also represents the minimum or maimum value of the parabola. o A maimum occurs when o A minimum occurs when The parabola will be: o concave up if o concave down if Every parabola has an ais of symmetry which is the line that runs down its 'center. o The ais of symmetry runs through the o The equation for the ais of symmetry is:
7 Finding Key Points from a Graph: Eample: For each graph, find the -intercepts and y-intercept and the verte. Is the verte a maimum or a minimum? Draw a dotted line for the ais of symmetry and write the equation for the ais of symmetry. y-intercept: y-intercept: y-intercept: y-intercept: -intercepts: -intercepts: -intercepts: -intercepts: verte: verte: verte: verte: MAX or MIN MAX or MIN MAX or MIN MAX or MIN ais of symmetry: ais of symmetry: ais of symmetry: ais of symmetry: Eample: Sketch the graph of the quadratic functions given the key features. Label the key features on your graph. Draw a dotted line for the ais of symmetry. 1. -intercepts: 1, 0 & 5, 0. -intercept: -, 0. -intercept: none y-intercept: 0, 10 y-intercept: 0, -6 y-intercept: 0, 4 minimum:, -8 maimum: -, 0 minimum:, You are presented with the graph below. How could you find the eact line of the ais of symmetry? Will this method always work?
8 Finding Key Points from a Table Find the minimum point verte of the function given a table of values. 1. Look for the lowest y-value.. Check if the lowest y-value is evenly sandwiched between the same value. Find the maimum point verte of the function given a table of values. 1. Look for the highest y-value.. Check if the highest y-value is evenly sandwiched between the same value. For the following eamples, use the table to identify the and y intercepts and the verte. Is the verte a maimum or a minimum? Write an equation for the ais of symmetry y y y y-intercept: y-intercept: y-intercept: -intercepts: -intercepts: -intercepts: verte: verte: verte: MAX or MIN MAX or MIN MAX or MIN concave up or down? concave up or down? concave up or down? ais of symmetry: ais of symmetry: ais of symmetry: Finding Key Points from the Equation When given an equation, we can find the key points algebraically. We will discuss -intercepts net unit but for now: To find the y-intercept, we To find the -intercepts, we Practice: Given the equation, find the y-intercept. 1. y = 9. y =. y = y = y-intercept: y-intercept: y-intercept: y-intercept:
9 NOTES: ALGEBRA 4. GRAPHING QUADRATICS BY HAND STARTER: For the following eamples, identify the key features of the quadratic. 1. f 5. y. -intercept: y-intercept: -intercept: y-intercept: verte: MAX or MIN: Concave up or down? Ais of Symmetry: -intercept: y-intercept: verte: MAX or MIN: y Concave up or down? Ais of Symmetry: 4. Given f 4, find f GRAPHING QUADRATIC FUNCTIONS We have talked about identifying key features from a graph, table, and equation. Today we will discuss graphing quadratic functions by hand from the equation. First, let s review standard form of a quadratic function: STANDARD FORM OF A QUADRATIC FUNCTION: f a bc where a, b, and c are integers and a 0. Eample: For the following functions, identify A, B, C. Is the quadratic function in standard form? 1. f 6 5. g. h 4 A= B= C= A= B= C= A= B= C= Standard form? Standard form? Standard form?
10 All parabolas have an ais of symmetry. If you were to fold a parabola along its ais of symmetry, the points on either side of this line would match. Eample: Find the ais of symmetry in the following quadratic functions. a. b. c. d. 1 4 h e. 5 y c. 1 h The maimum or minimum point on a parabola is called the verte. Eample: Find the verte in the following quadratic functions. a. 8 f b g c. y d. 1 4 h e. 5 y f. 1 h 8 f 15 8 g y AXIS OF SYMMETRY: The ais of symmetry is a vertical line that goes through the middle of the parabola and splits the parabola into equal parts. The parabola mirrors itself on either side of the ais of symmetry. a b VERTEX: The verte is where the ais of symmetry and the parabola intersect. The -coordinate is the ais of symmetry a b. To find the y-coordinate, plug in a b and solve for y.
11 STEPS TO GRAPHING A QUADRATIC IN STANDARD FORM: 1. Find the ais of symmetry by using = b a.. Identify the verte. Remember, the coordinate of the verte is the same as the ais of symmetry. Make a table and find points on either side of the verte. 4. Plot points. 5. Connect points, making sure your graph forms a parabola Eample: Complete the table and graph the function. Identify the ais of symmetry and the verte. State the domain and range. a. f = b. g 8 15 c. y + 4 5, y, y 8 15, y Ais of symmetry: Ais of symmetry: Ais of symmetry: Verte:, Verte:, Verte:, Domain = Range = Domain = Range = Domain = Range =
12 Maimum and minimum values: The y-coordinate of the verte is the maimum value or the minimum value. When the graph opens up, there is a minimum value. When the graph opens down, there is a maimum value. minimum maimum Eample: Determine whether the function has a maimum or minimum value. State the maimum or minimum value of each. a. f 8 b. g 8 15 c. y Eample: Fred jumps feet first from a diving board, springing up into the air and then dropping feet-first. The distance d in feet from his feet to the pool s surface t seconds after he jumps is d t 16t 18t. a Draw a picture of what Fred s jump would look like. b When does the maimum height occur? c What is the maimum height of Fred s feet during this jump? d What does the constant term in the equation tell you about Fred s jump?
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