Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
Writing Function Formulas for Quadratic Graphs
In the last two lessons, you practiced making graphs when given a quadratic function formula.
The goal of today’s lesson is to learn to do the reverse: write a quadratic function formula when
given a graph.
Steps for writing a quadratic function formula
(1) Choose a function form that fits the information that’s easiest to get from the graph.
 If you can read the coordinates of the vertex easily (such as when they are whole numbers),
you can pick vertex form: f(x) = a(x – h)2 + k.
 If the graph has two zeros and they are easy to read from the graph (such as when they are
whole numbers), you can use a factored form like this: f (x)  a(x  x1)(x  x2 )
 Sometimes both the zeros and the vertex are easy to get, so either method could be used.
(2) Put in numbers using the known information.

 If the zeros are known, put them in as x1 and x2 in the formula f (x)  a(x  x1)(x  x2 )
 If the vertex coordinates are known, put them in as h and k in the formula f(x) = a(x – h)2 + k.
After step (2), the only work that remains is to find the value of a.



(3) Put in an input-output pair, then solve for a.
 Pick any single point from the graph that you can read easily (such as a point with whole
number coordinates), other than the zeros or vertex you’ve already used.
Substitute its coordinates in place of x and f(x).
 Solve the equation for a.
(4) Write your answer.
 Just take your equation from step (2) and put in the a found in step (3).
Examples
Example 1: Find the function formula for the given parabola.
This function doesn’t have any zeros, so vertex form is our
only option.
f(x) = a (x – 1)2 + 3
6 = a (4 – 1)2 + 3
6 =a· 9
+3
3 =a· 9
1
3 = a
Answer: f(x) = 13 (x – 1)2 + 3
Use the vertex to write:
Picking the point (4, 6), substitute:
Solve for a:
Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
Example 2: Find the function formula for the given parabola.
This function’s vertex is off the top of the grid, so factored
form is our only option.
f(x) = a (x + 4)(x – 1)
8 = a (0 + 4)(0 – 1)
8 = a · 4 · (–1)
–2 = a
Answer: f(x) = –2 (x + 4)(x – 1)
Use the zeros to write:
Picking the point (0, 8), substitute:
Solve for a:
You try it
1. Find quadratic function formulas for these parabolas. Use vertex form and the method shown
in Example 1.
a.
b.
Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
2. Find quadratic function formulas for these parabolas. Use factored form and the method
shown in Example 2.
a.
b.
Homework Problems
3. Two quadratic functions are shown in the graph. Find
function formulas for both of them. Tell what is alike
and what is different about the two formulas.
Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
4. Two quadratic functions are shown in the graph. Find
function formulas for both of them. Tell what is alike
and what is different about the two formulas.
5. a. Find a function formula in vertex form for the
given graph (use the method from Example 1).
b. Find a function formula in factored form for the
given graph (use the method from Example 2).
c. Algebraically show that the answers to parts a and b agree with each other.
Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
6. Find quadratic function formulas for these parabolas. Only one of the methods works well for
each problem.
a.
b.
c.
d.
Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
7. In each part, find a function formula for a quadratic function f(x) that fits the given
information. The information is comparable to what you took from the graphs in other
problems, so you can still use the method of either Example 1 or Example 2.
a. f(x) has vertex (2, 5) and one of its other points is (3, 9).
(Use the method of Example 1.)
b. f(x) has zeros at x = 1 and x = 7 and one of its other points is (–1, 1).
(Use the method of Example 2.)
c. f(x) has the following points on its graph: (–2, 0), (2, 0), and (3, –4).
Name:
Date: _________________
Algebra 2
Writing Function Formulas for Quadratic Graphs
d. f(x) has f(2) = 3 as its minimum value, and one of its other values is f(1) = 8.
e. f(x) has the following points as its x- and y-intercepts: (0, –6), (3, 0), and (5, 0).
f. f(x) describes the height of a ball in the air, measured in feet, x seconds after it was
thrown. When the ball was thrown (x = 0) its height was 4 feet. It takes 2 seconds for the
ball to reach its greatest height, 68 feet.