10.3

10.3 Graph

Characteristics ofThe general form where

is called the intercept form of a quadratic function.

The x-intercepts are p and q.The axis of symmetry is halfway between

and AoS: The parabola opens up if and down if

Example 1 Find x-intercepts

Find the x-intercepts of the graph of ( )1+x= ( ).5x–y –

SOLUTION

To find the x-intercepts, you need to find the values of xwhen y = 0.

Write original function.=y ( )1+x ( )5x– –

Substitute 0 for y.=0 ( )1+x ( )5x– –

Zero-product property1+x 5x – = 0= 0 or

Solve for x.x 5x == or1–

The x-intercepts are and 5.1–ANSWER

Example 2 Graph a quadratic function in intercept form

Graph ( )1 +x= ( ).3x–y –2

STEP 2Find and draw the axis of symmetry:

2

q+px = =2+1 ( )3–

= 1.–

STEP 1Identify the x-intercepts. The x-intercepts are and Plot (1, 0) and ( 3, 0).

p = 1q = –3. –

Example 2 Graph a quadratic function in intercept form

STEP 4Draw a parabola through the vertex and the points where the x-intercepts occur.

So, the vertex is ( , 8).–1

STEP 3Find and plot the vertex. The axis of symmetry is x , so the x-coordinate of the vertex is . To find the y-coordinate of the vertex, substitute for x and simplify.

= –1–1

–1

( )1 += ( )3–y –2 1– 1– 8=

Example 3 Graph a quadratic function in standard form

Graph y = 12 +x2 12.x–3

Rewrite the quadratic function in intercept form.STEP 1

Write the function.=y 12 +x2 12x–3

Factor out 3.=y 4 +x2 4x–3( )

= x 2–3( )2 Factor the trinomial.

Write in intercept form.= x 2–3( ) x 2–( )

STEP 2Identify the x-intercepts. There is one x-intercept, 2. Plot (2, 0).

Example 3

STEP 3Find and draw the axis of symmetry:

2

q+px = = = 2.2

2+2

STEP 4Find and plot the vertex. The axis of symmetry is x 2, so the x-coordinate of the vertex is 2, which is also the x-intercept. So, the vertex is (2, 0).

=

Graph a quadratic function in standard form

Example 3

Draw a parabola through the points.STEP 6

STEP 5Plot a point and its reflection. Choose a value for x, say x 1. When x 1, y 3. Plot (1, 3). By reflecting the point in the axis of symmetry, you can also plot (3, 3).

= = =

Graph a quadratic function in standard form

Example 4 Write a quadratic function in intercept form

Write a quadratic function in intercept form whose graph has x-intercepts 1 and 3 and passes through the point (0, 12).

–

–

STEP 1

Substitute the x-intercepts into

The x-intercepts are p

( )px= ( ).y –a qx –

and= 1– q 3.=

( )1x= ( )y +a 3x – Simplify.

–Substitute 1 for p and 3 for q.( x= ( )y –a 3x –( )1– )

Example 4 Write a quadratic function in intercept form

Find the value of a in using the given point (0, 12).

STEP 2

( )1x= (y +a 3x – )

–

Simplify.12– = 3a–

Divide each side by 3.–4 = a

ANSWER

y =The function in intercept form is ( )1x (+4 3x – ).

Substitute 0 for x and 12 for y.–( )10= ( )+ 30 –12– a

Example 5 Model a parabolic path using intercept form

BIOLOGY

When a dolphin leaps out of the water, its body follows a parabolic path through the air. Write a function whose graph is the path of the dolphin in the air.

SOLUTION

STEP 1

( )0x= (y a 4x – ),– = (y ax 4x – ).or

Identify the x-intercepts. The dolphin leaves the water at (0, 0) and re-enters at (4, 0), so the x-intercepts of the path are p 0 and q 4. The function is of the form

= =


STEP 2

Find the axis of symmetry and the vertex. The axis of symmetry is

The maximum height of 2 meters occurs on the axis of symmetry,

so the vertex of the graph is (2, 2).24+0x = 2.=

Find the value of a. Substitute the coordinates of

the vertex into the function. The vertex is (2, 2), so

2 a (2 – 0)(2 – 4), or a

STEP 3

= =2

1– .


ANSWER

The graph of the functionpath.

is the dolphin’s2

1= – (x 4x – )y

10.3 Warm-UpFind the x-intercepts of the graph of the quadratic function. 1. Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercept(s).2. Write a quadratic function in intercept form whose graph has the given x-intercept(s) and passes through the given point.3. x-intercepts: -6 and 2. point:

Education

10.3