1.1 Graphing Quadratic Functions (p. 2) Definitions Standard form of quad. function Steps for graphing Minimums and maximums

Transcription

1 1.1 Graphing Quadratic Functions (p. 2) Definitions Standard form of quad. function Steps for graphing Minimums and maximums

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3 Quadratic Function A function of the form y=ax 2 +bx+c where a 0 making a u-shaped graph called a parabola. Example quadratic equation:

4 Vertex- The lowest or highest point of a parabola. This is the maximum or minimum of the graph. Vertex Axis of symmetry- Axis of Symmetry The vertical line through the vertex of the parabola.

5 Standard Form Equation y=ax 2 + bx + c If a is positive, u opens up If a is negative, u opens down The x-coordinate of the vertex is at: -b/2a To find the y-coordinate of the vertex, plug the x-coordinate into the given eqn. The axis of symmetry is the vertical line x= -b/2a Choose 2 x-values on either side of the vertex x-coordinate. Use the eqn to find the corresponding y-values. Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

6 Example 1 Graph a function of the form y = ax 2 Graph y = 2x 2. Compare the graph with the graph of y = x 2. First, find the vertex. SOLUTION STEP 1 STEP 2 STEP 3 Plot the points from the table. Draw a smooth curve through the points.

7 Graph a function of the form y = ax 2 STEP 4 Compare the graphs of y = 2x 2 and y = x 2. Both open up and have the same vertex and axis of symmetry. The graph of y = 2x 2 is narrower than the graph of y = x 2.

8 Example 2: Graph a function of the form y = ax 2 + c graph of y = x 2 SOLUTION STEP 1 Make a table of values for y = 1 x STEP 2 Plot the points from the table. STEP 3 Draw a smooth curve through the points.

9 Example 2 continued STEP x 2 + 3

10 Practice y = x 2 5 SOLUTION STEP 1 Make a table of values for y = x 2 5. X Y STEP 2 STEP 3 Plot the points from the table. Draw a smooth curve through the points. STEP 4 Compare the graphs of y = x 2 5 and y = x 2.

11 Practice Answer ANSWER Same axis of symmetry, vertex is shifted down 5 units, and opens down

12 Example 3: Graph y=2x 2-8x+6 a=2 Since a is positive the parabola will open up. Vertex: use b=-8 and a=2 Vertex is: (2,-2) x=2 Axis of symmetry is the vertical line x=2 Table of values for other points: x y * Graph!

13 Now you try one! y=x 2 2x 1 * Open up or down? * Vertex? * Axis of symmetry?

14 Graph the function. Label the vertex and axis of symmetry. y = x 2 2x 1 SOLUTION STEP 1 STEP 2 Identify the coefficients of the function. The coefficients are a = 1, b = 2, and c = 1. Because a > 0, the parabola opens up. Then find the x - coordinate of the vertex. ( 2) x = - b = 2(1) = 1 2a Find the vertex. Calculate the y - coordinate. y = = 2

15 Practice answer So, the vertex is (1, 2). Plot this point. STEP 3 Draw the axis of symmetry x = 1. STEP 4 Select 4 more points to plot. 2 on each side of the axis of symmetry. Lets try x = -1, x = 0, x = 2, x = 3

16 Find the minimum or maximum value Tell whether the function y = 3x 2 18x + 20 has a minimum value or a maximum value. Then find the minimum or maximum value. SOLUTION Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex. x = b 18) = ( 2a 2(3) = 3 y = 3(3) 2 18(3) + 20 = 7 ANSWER The minimum value is y = 7. You can check the answer on a graphing calculator.

17 Solve a multi-step problem Go - Carts A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for every $1 reduction in the price per racer. How can the owner of the go-cart track maximize weekly revenue?

18 SOLUTION STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the weekly revenue. STEP 2 Write a verbal model. Then write and simplify a quadratic function. R(x) = 13, x 380x 20x 2 R(x) = 20x x + 13,300 STEP 3 Find the coordinates (x, R(x)) of the vertex. Evaluate R(8). Find x - coordinate. x = b 2a = 320 2( 20) = 8 R(8) = 20(8) (8) + 13,300 = 14,580

19 ANSWER The vertex is (8, 14,580), which means the owner should reduce the price per racer by $8 to increase the weekly revenue to $14,580.

20 What If? In Example 5, suppose each $1 reduction in the price per racer brings in 40 more racers per week. How can weekly revenue be maximized? SOLUTION STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the weekly revenue. STEP 2 Write a verbal model. Then write and simplify a quadratic function. R(x) = 20x x + 13,300

21 STEP 3 Find the coordinates (x, R(x)) of the vertex. Find x - coordinate. Then, evaluate R(12.75). R(12.75) = 40(12.75) (12.75) + 13,300 = ANSWER The vertex is (12.75, 19,802.5), which means the owner should reduce the price per racer by $12.75 to increase the weekly revenue to $19,

22 Assignment p even, 29, 30, 32, 34 For graphing problems: Does the function have a max or min?