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. SOLUTION STEP 1 Make a table of values for y = 2x 2. STEP 2 Plot the points from the table. STEP 3 Draw a smooth curve through the points.

EXAMPLE 1 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.

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

EXAMPLE 2 Graph a function of the form y = ax 2 STEP 4 Compare the graphs of y = – and y = x 2. Both graphs have the same axis of symmetry. However, the graph of y = – opens down and is wider than the graph of y = x 2. Also, its vertex is 3 units higher. x

EXAMPLE 3 Graph a function of the form y = ax 2 + bx + c Graph y = 2x 2 – 8x + 6. SOLUTION Identify the coefficients of the function. The coefficients are a = 2, b = – 8, and c = 6. Because a > 0, the parabola opens up. STEP 1 STEP 2 Find the vertex. Calculate the x - coordinate. x = b2a b2a = (– 8) 2(2) –– = 2 Then find the y - coordinate of the vertex. y = 2(2) 2 – 8(2) + 6 = – 2 So, the vertex is (2, – 2). Plot this point.

EXAMPLE 3 Graph a function of the form y = ax 2 + bx + c STEP 3Draw the axis of symmetry x = 2. STEP 4Identify the y - intercept c, which is 6. Plot the point (0, 6). Then reflect this point in the axis of symmetry to plot another point, (4, 6). STEP 5 Evaluate the function for another value of x, such as x = 1. y = 2(1) 2 – 8(1) + 6 = 0 Plot the point (1, 0) and its reflection (3, 0) in the axis of symmetry.

EXAMPLE 3 Graph a function of the form y = ax 2 + bx + c STEP 6 Draw a parabola through the plotted points.

EXAMPLE 4 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 = – b2a b2a = – (– 18) 2a = 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.

EXAMPLE 5 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 ?

EXAMPLE 5 Solve a multi-step problem 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

EXAMPLE 5 Solve a multi-step problem STEP 3 Find the coordinates (x, R(x)) of the vertex. x = – b2a b2a = – 320 2(– 20) = 8 Find x - coordinate. R(8) = – 20(8) (8) + 13,300 = 14,580 Evaluate R(8). 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.

EXAMPLE 1 Graph a quadratic function in vertex form Graph y = – (x + 2) SOLUTION STEP 1 Identify the constants a = –, h = – 2, and k = 5. Because a < 0, the parabola opens down STEP 2 Plot the vertex (h, k) = ( – 2, 5) and draw the axis of symmetry x = – 2.

EXAMPLE 1 Graph a quadratic function in vertex form STEP 3 Evaluate the function for two values of x. x = 0: y = (0 + 2) = – x = 2: y = (2 + 2) = – Plot the points (0, 4) and (2, 1) and their reflections in the axis of symmetry. STEP 4 Draw a parabola through the plotted points.

EXAMPLE 2 Use a quadratic model in vertex form Civil Engineering The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable can be modeled by the function. y = (x – 1400) where x and y are measured in feet. What is the distance d between the two towers ?

EXAMPLE 2 Use a quadratic model in vertex form SOLUTION The vertex of the parabola is (1400, 27). So, a cable’s lowest point is 1400 feet from the left tower shown above. Because the heights of the two towers are the same, the symmetry of the parabola implies that the vertex is also 1400 feet from the right tower. So, the distance between the two towers is d = 2 (1400) = 2800 feet.

EXAMPLE 3 Graph a quadratic function in intercept form Graph y = 2(x + 3) (x – 1). SOLUTION STEP 1 Identify the x - intercepts. Because p = – 3 and q = 1, the x - intercepts occur at the points (– 3, 0) and (1, 0). STEP 2 Find the coordinates of the vertex. x = p + q 2 – = – 1= y = 2(– 1 + 3)(– 1 – 1) = – 8

EXAMPLE 3 Graph a quadratic function in intercept form STEP 3 Draw a parabola through the vertex and the points where the x - intercepts occur.

EXAMPLE 4 Use a quadratic function in intercept form Football The path of a placekicked football can be modeled by the function y = – 0.026x(x – 46) where x is the horizontal distance (in yards) and y is the corresponding height (in yards). a. How far is the football kicked ? b. What is the football’s maximum height ?

EXAMPLE 4 Use a quadratic function in intercept form SOLUTION a. Rewrite the function as y = – 0.026(x – 0)(x – 46). Because p = 0 and q = 46, you know the x - intercepts are 0 and 46. So, you can conclude that the football is kicked a distance of 46 yards. b. To find the football’s maximum height, calculate the coordinates of the vertex. x = p + q = 23= y = – 0.026(23)(23 – 46) 13.8 The maximum height is the y-coordinate of the vertex, or about 13.8 yards.

EXAMPLE 5 Change from intercept form to standard form Write y = – 2 (x + 5) (x – 8) in standard form. y = – 2 (x + 5) (x – 8) Write original function. = – 2 (x 2 – 8x + 5x – 40) Multiply using FOIL. = – 2 (x 2 – 3x – 40) Combine like terms. = – 2x 2 + 6x + 80 Distributive property

EXAMPLE 6 Change from vertex form to standard form Write f (x) = 4 (x – 1) in standard form. f (x) = 4(x – 1) Write original function. = 4(x – 1) (x – 1) + 9 = 4(x 2 – x – x + 1) + 9 Multiply using FOIL. Rewrite (x – 1) 2. = 4(x 2 – 2x + 1) + 9 Combine like terms. = 4x 2 – 8x Distributive property = 4x 2 – 8x + 13 Combine like terms.