10.2

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Graph y = ax2 + bx + c

10.2 Warm-Up

Evaluate the expression.

1. x2 – 2 when x = 3

2. 2x2 + 9 when x = 2

ANSWER 7

ANSWER 17

ANSWER 6.25 in.2

Martin is replacing a square patch of counter top. The area of the patch is represented by A = s2. What is the area of the patch if the side length is 2.5 inches?

3.

10.2 Example 1

Consider the function y = –2x2 + 12x – 7.

a. Find the axis of symmetry of the graph of the function.

b. Find the vertex of the graph of the function.

SOLUTION

122(– 2)x = – b

2a= = 3 Substitute –2 for a and 12 for b.

Then simplify.

For the function y = –2x2 + 12x – 7, a = 2 and b = 12.a.

10.2 Example 1

b. The x-coordinate of the vertex is , or 3. b 2a

–

y = –2(3)2 + 12(3) – 7 = 11 Substitute 3 for x. Then simplify.

ANSWER The vertex is (3, 11).

10.2 Guided Practice

1. Find the axis of symmetry and vertex of the graph of the function y = x2 – 2x – 3.

ANSWER x = 1, (1, –4).

10.2 Example 2

Graph y = 3x2 – 6x + 2.

Determine whether the parabola opens up or down. Because a > 0, the parabola opens up.

STEP 1

STEP 2 =Find and draw the axis of symmetry: x = – b

2a– – 62(3)= 1.

STEP 3Find and plot the vertex.

To find the y-coordinate, substitute 1 for x in the function and simplify. y = 3(1)2 – 6(1) + 2 = – 1

So, the vertex is (1, – 1).

The x-coordinate of the vertex is b2a

, or 1.–

10.2 Example 2

STEP 4Plot two points. Choose two x-values less than the x-coordinate of the vertex. Then find the corresponding y-values.

STEP 5

Reflect the points plotted in Step 4 in the axis of symmetry.

STEP 6

Draw a parabola through the plotted points.

10.2 Guided Practice

2. Graph the function y = 3x2 + 12x – 1. Label the vertex and axis of symmetry.

ANSWER

10.2 Example 3

Tell whether the function f(x) = – 3x2 – 12x + 10 has aminimum value or a maximum value. Then find theminimum or maximum value.

SOLUTION

Because a = – 3 and – 3 < 0, the parabola opens down andthe function has a maximum value. To find the maximumvalue, find the vertex.

x = – = – = – 2 b2a

– 122(– 3)

f(–2) = – 3(–2)2 – 12(–2) + 10 = 22 Substitute –2 for x. Thensimplify.

The x-coordinate is – b2a

ANSWER

The maximum value of the function is f(– 2) = 22.

10.2 Guided Practice

3. Tell whether the function f(x) = 6x2 + 18x + 13 has aminimum value or a maximum value. Then find theminimum or maximum value.

1 2

Minimum value;

ANSWER

10.2 Example 4

The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled by the graph of y = 0.000097x2 – 0.37x + 549 where x and y are measured in feet. What is the height of the cable above the water at its lowest point?

SUSPENSION BRIDGES

10.2 Example 4

SOLUTION

The lowest point of the cable is at the vertex of theparabola. Find the x-coordinate of the vertex. Use a = 0.000097 and b = – 0.37.

x = – = – ≈ 1910 b2a

– 0.372(0.000097)

Use a calculator.

Substitute 1910 for x in the equation to find they-coordinate of the vertex.

y ≈ 0.000097(1910)2 – 0.37(1910) + 549 ≈ 196

ANSWER

The cable is about 196 feet above the water at its lowest point.

10.2 Guided Practice

SUSPENSION BRIDGES

4. The cables between the two towers of the Takoma Narrows Bridge form a parabola that can be modeled by the graph of the equation y = 0.00014x2 – 0.4x + 507 where x and y are measured in feet. What is the height of the cable above the water at its lowest point? Round your answer to the nearest foot.

ANSWER

221 feet

10.2 Lesson Quiz

1. Find the axis of symmetry and the vertex of the graph of the function y = – 4x2 + 8x – 9

2. Graph y = –2x2 + 4x + 1

ANSWER x = 1; (1, – 5)

ANSWER

10.2 Lesson Quiz

3. An arch to the entrance of the library can be modeled by y = – 0.13x2 + 2.5x, where x and y are measured in feet. To the nearest foot, what is the height of the highest point of the arch?

ANSWER 12 ft