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In this section, we are going to translate a graph into an algebraic equation.

man juggling 2 balls labeled graph and algebraic equation

chart showing two methods to use to write an algebraic equation from a graph – Method 1 is use the zeros and any other point on the graph – Method 2 is use the vertex and any other point on the graph

Using Method 1

Write an algebraic equation for the parabola shown in the graph.

graph of a parabola opening up

First,we need to find the zeros (x-intercepts) and one other point on the graph.

By looking at the graph, the zeros are at x = -5 and 1.

If x = -5 is a zero, then (x + 5) is a factor.
If x = 1 is a zero, then (x – 1) is a factor.

Using the factored form for a quadratic function, y,we get

y = a(x+5)(x – 1)

Next, pick any point on the graph, like (-3, -16).

Substitute the coordinates of this point in for x and y and solve for a.

y = a(x+5)(x – 1)
-16 = a(-3+5)(-3 – 1)
-16 = a(2)(-4)
-16 = -8a
a = 2

** Notice that if you had picked a different point like (0, -10), “a” would still equal 2. So, any point on the graph will work.

Now, substitute “a” back into the factored form from the first step and simplify.

y = a(x+5)(x – 1)
y = 2(x+5)(x – 1)

y = 2x² + 8x – 10

Now let’s use Method 2 to find the equation.

graph of a parabola opening up

First, we need to find the vertex.

The vertex is at (-2, -18), so h = -2 and k = -18.
Substituting into the vertex form, y = a(x – h)² + k, for a parabola we get:

y = a(x – h)² + k
y = a(x – -2)² + -18
y = a(x + 2)² – 18

Next, pick any point on the graph, like (-3, -16).

Substitute the coordinates of this point in for x and y above and solve for “a”.

y = a(x + 2)² – 18
-16 = a(-3 + 2)² – 18
-16 = a(-1)² – 18
-16 = a – 18

a = 2

**Just like in Method 1, if you had picked a different point like (0, -10), “a” would still equal 2. So, any point on the graph will work.

Now, substitute “a” back into the vertex form and simplify.
y = a(x + 2)² – 18
y = 2(x + 2)² – 18
y= 2(x+2)(x+2) – 18
y= 2(x² + 4x + 4) – 18
y= 2x² + 8x – 10

Observe that both methods ended up with the same equation as the answer.

Click here for extra practice.