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Draw the axis of symmetry, x h 2. = Example 1 Graph a Quadratic Function in Vertex Form Graph . ( )2 2 x – y = 1 + SOLUTION The function is in vertex form where , h 2, and k 1. Because a < 0, the parabola opens down. = a 2 – y k + ( )2 h x STEP 1 Draw the axis of symmetry, x h 2. = STEP 2 Plot the vertex, . ( ) h, k 2, 1 = STEP 3 Plot points. The x-values 3 and 4 are to the right of the axis of symmetry.
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Warm-Up Exercises
Find the product.
1. x + 6(
) x + 3(
) ANSWER x 2 18+ 9x +
2. x – 5(
)2 ANSWER x 2 25+10x–
ANSWER 4x 2 100–x + 5(
)3. x – 5(
)4
4. Write y x 8x 12 5 in standard form.= +(
) +
ANSWER 8x 2 5+ 12x +
Example 1 Graph a Quadratic Function in Vertex Form
Graph .( )22x –y = 2– 1+
SOLUTIONThe function is in vertex form where , h 2, and k 1. Because a < 0, the parabola opens down.
=a = 2–y = a k+( )2hx –
=
STEP 1 Draw the axis of symmetry, x h 2. = =
STEP 2 Plot the vertex, .( )h, k ( )2, 1=
STEP 3 Plot points. The x-values 3 and 4 are to the right of the axis of symmetry.
Example 1 Graph a Quadratic Function in Vertex Form
( )223 –y = 2– 1+ = 1–
( )224 –y = 2– 1+ = 7–
One point on the parabola is .( )3, –1
STEP 4 Draw a parabola through the points.
Plot the points and . Then plot their mirror images across the axis of symmetry.
( )3, –1 ( )4, –7
Another point on the parabola is ( )4, –7 .
Example 2 Graph a Quadratic Function in Intercept Form
Graph .( )3x –( )1x +y =
SOLUTIONThe function is in intercept form where a 1, , and q 3. Because a > 0, the parabola opens up.
= p = 1– =y = ( )qx –( )px –a
STEP 1 Draw the axis of symmetry. The axis of symmetry is:
x =qp
2+
=31
2+–
1=
Example 2 Graph a Quadratic Function in Intercept Form
STEP 2 Find and plot the vertex.The x-coordinate of the vertex is x 1. Calculate the y-coordinate of the vertex.
=
( )3x –( )1x +y =
( )31 –( )11 += = 4–
Plot the vertex . ( )1, –4
STEP 3 Plot the points where the x-intercepts occur. The x-intercepts are and q 3. Plot the points and .
=p = 1–( )1, 0– ( )3, 0
Example 2 Graph a Quadratic Function in Intercept Form
STEP 4 Draw a parabola through the points.
Checkpoint Graph a Quadratic Function
Graph the function. Label the vertex and the axis of symmetry.
ANSWER
1. ( )23x –y = 1–
Checkpoint Graph a Quadratic Function
Graph the function. Label the vertex and the axis of symmetry.
ANSWER
2. 3y = ( )22x –– +
Checkpoint Graph a Quadratic Function
Graph the function. Label the vertex and the axis of symmetry.
ANSWER
3. 4y = ( )21x2 ++
Checkpoint Graph a Quadratic Function
Graph the function. Label the vertex and the x-intercepts.
ANSWER
4. ( )7x –y = ( )3x –
Checkpoint Graph a Quadratic Function
Graph the function. Label the vertex and the x-intercepts.
ANSWER
5. ( )5x –( )2x –y = –
Checkpoint Graph a Quadratic Function
Graph the function. Label the vertex and the x-intercepts.
ANSWER
6. ( )3x –y = ( )1x +2
Example 3 Find the Minimum or Maximum Value
Tell whether the function has a minimum value or a maximum value. Then find the minimum or maximum value.
( )4x –( )6x +y = 4–
SOLUTIONThe function is in intercept form where a 4, p 6, and q 4. Because a < 0, it has a maximum value. Find the y-coordinate of the vertex.
( )qx –y = a( )px ––= –= =
( )4x –( )6x +y = 4– = 4– ( )61 +– ( )41– – = 100
x =qp
2+
=46
2+– 1–=
Example 3 Find the Minimum or Maximum Value
The maximum value of the function is 100.ANSWER
Example 4 Using a Quadratic Function
Civil Engineering The Golden Gate Bridge in San Francisco, CA, has two towers. The top of each tower is 500 feet above the road. The towers are connected by suspension cables. Each cable forms a parabola with the equation where x and y are measured in feet.
y = 818690
–( )22100x +
Find the height of the cable above the road, when the cable is at the lowest point. The road is represented by y 0.
a.
=
Example 4 Using a Quadratic Function
What is the distance d between the towers? b.
SOLUTION
The function is in vertex form where .a =
18690
2100,, h = and k = 8
The vertex is . The height of the cable at its lowest point is the y-coordinate of the vertex. So, the cable is 8 feet above the road.
a. ( )h, k ( )2100, 8=
Example 4 Using a Quadratic Function
The vertex is 2100 feet from the left tower. The axis of symmetry passes through the vertex. So, the vertex is also 2100 feet from the right tower. Thedistance between the towers is 2100 2100 4200 feet.
b.
=+
Checkpoint Find the Minimum or Maximum Value
Tell whether the function has a minimum or maximum value. Then find the minimum or maximum value.
8. ( )7x –y = 3( )4x –
7. ( )28xy = 12–21
+ ANSWER minimum; 12–
ANSWER minimum; –427
( )4x +y = x9. ANSWER minimum; 4–
