1. Write a quadratic function in vertex form
EXAMPLE 1
Write a quadratic function in vertex form
Write a quadratic function for the parabola shown.
SOLUTION
Use vertex form because the vertex is given.
y = a(x – h)2 + k
Vertex form
y = a(x – 1)2 – 2
Substitute 1 for h and –2 for k.
Use the other given point, (3, 2), to find a.
2 = a(3 – 1)2 – 2
Substitute 3 for x and 2 for y.
2 = 4a – 2
Simplify coefficient of a.
1 = a
Solve for a.

2. EXAMPLE 1
Write a quadratic function in vertex form
ANSWER
A quadratic function for the parabola is y = (x – 1)2 – 2.

3. Write a quadratic function in intercept form
EXAMPLE 2
Write a quadratic function in intercept form
Write a quadratic function for the parabola shown.
SOLUTION
Use intercept form because the x-intercepts are given.
y = a(x – p)(x – q)
Intercept form
y = a(x + 1)(x – 4)
Substitute –1 for p and 4 for q.

4. Write a quadratic function in intercept form
EXAMPLE 2
Write a quadratic function in intercept form
Use the other given point, (3, 2), to find a.
2 = a(3 + 1)(3 – 4)
Substitute 3 for x and 2 for y.
2 = –4a
Simplify coefficient of a. 1 2 – = a
Solve for a.
A quadratic function for the parabola is 1 2 –
(x + 1)(x – 4) .
y =
ANSWER

5. EXAMPLE 3
Write a quadratic function in standard form
Write a quadratic function in standard form for the parabola that passes through the points (–1, –3), (0, –4), and (2, 6).
SOLUTION
STEP 1
Substitute the coordinates of each point into y = ax2 + bx + c to obtain the system of three linear equations shown below.

6. Write a quadratic function in standard form
EXAMPLE 3
Write a quadratic function in standard form
–3 = a(–1)2 + b(–1) + c
Substitute –1 for x and 23 for y.
–3 = a – b + c
Equation 1
–3 = a(0)2 + b(0) + c
Substitute 0 for x and –4 for y.
–4 = c
Equation 2
6 = a(2)2 + b(2) + c
Substitute 2 for x and 6 for y.
6 = 4a + 2b + c
Equation 3
STEP 2
Rewrite the system of three equations in Step 1 as a system of two equations by substituting –4 for c in Equations 1 and 3.

7. Write a quadratic function in standard form
EXAMPLE 3
Write a quadratic function in standard form
a – b + c = –3
Equation 1
a – b – 4 = –3
Substitute –4 for c.
a – b = 1
Revised Equation 1
4a + 2b + c = 6
Equation 3
4a + 2b – 4 = 6
Substitute –4 for c.
4a + 2b = 10
Revised Equation 3
STEP 3
Solve the system consisting of revised Equations 1 and 3. Use the elimination method.

8. EXAMPLE 3
Write a quadratic function in standard form
a – b = 1
2a – 2b = 2
4a + 2b = 10
4a + 2b = 10
6a = 12
a = 2
So 2 – b = 1, which means b = 1.
The solution is a = 2, b = 1, and c = –4.
A quadratic function for the parabola is y = 2x2 + x – 4.
ANSWER

9. GUIDED PRACTICE
for Examples 1, 2 and 3
Write a quadratic function whose graph has the given characteristics.
1. vertex: (4, –5)
passes through: (2, –1)
y = (x – 4)2 – 5
ANSWER
2. vertex: (–3, 1)
passes through: (0, –8)
ANSWER
y = (x + 3)2 + 1
y = (x + 2)(x – 5) 1 4
ANSWER
3. x-intercepts: –2, 5
passes through: (6, 2)

10. GUIDED PRACTICE
for Examples 1, 2 and 3
Write a quadratic function in standard form for the parabola that passes through the given points.
4. (–1, 5), (0, –1), (2, 11)
y = 4x2 – 2x – 1
ANSWER
5. (–2, –1), (0, 3), (4, 1) –5 12 7 6
ANSWER
y = x x + 3.
6. (–1, 0), (1, –2), (2, –15)
y = 4x2  x + 3
ANSWER