Quadratic equations are mathematical functions where one of the x variables is squared, or taken to the second power like this: x2. When these functions are graphed, they create a parabola which looks like a curved "U" shape on the graph. This is why a quadratic equation is sometimes called a parabola equation.
Two important values concerning these mathematical functions are the x-intercept and the y-intercept. The x-intercept indicates where the parabola graph of that function crosses the x axis. There can be one or two x intercepts for a single quadratic equations.
The y-intercept indicates where the parabola crosses the y axis. There is only one y intercept for each quadratic equation.
What Is the y Intercept of a Quadratic Function?
The y-intercept is where the parabola of a function crosses (or intercepts) the y axis. Another way to define the y-intercept is the value of y when x is equal to zero.
Because the y intercept is a point on a graph, you'll usually write it in point/coordinate form. For example, let's say your y value of the y intercept is 6.5. You would write the y intercept as (0, 6.5).
Different Forms of Quadratic Equations
Quadratic equations come in three general forms. These are the standard form, vertex form and factored form.
Standard form looks like this:
y = ax2 + bx + c where a, b and c are known constants and x and y are variables.
Vertex form looks like this:
y = a(x + b)2 + c where a, b and c are known constants and x and y are variables.
Factored form looks like this:
y = a(x + r1)(x + r2) where a is a known constant, r1 and r2 are "roots" of the equation (x intercepts), and x and y are variables.
Each of the forms looks drastically different, but the method for finding the y intercept of a quadratic equation is the same despite the various forms.
How to Find the Y Intercept of a Quadratic in Standard Form
Standard form is perhaps the most common and the easiest to understand. Simply plug zero (0) in as the value of x in the standard quadratic equation and solve. Here's an example.
Let's say your function is y = 5x2 + 11x + 72. Assign "0" as your x value and solve.
y = 5(0)2 + 11(0) + 72 = 72
You would then write the answer in the coordinate form of (0, 72).
How to Find the Y Intercept of a Quadratic in Vertex Form
As with standard form, simply plug "0" in as the value of x and solve. Here's an example.
Let's say your function is y = 134(x + 56)2 - 47. Assign "0" as your x value and solve.
y = 134(0 + 56)2 - 47 = 134(0)2 - 47 = -47
You would then write the answer in the coordinate form of (0, -47).
How to Find the Y Intercept of a Quadratic in Factored Form
Lastly, you have factored form. Again, you simply plug "0" in as the value of x and solve. Here's an example.
Let's say your function is y = 7(x - 8)(x + 2). Assign "0" as your x value and solve.
y = 7(0-8)(0+2) = 7(-8)(2) = -112
You would then write the answer in the coordinate form of (0, -112).
A Quick Trick
With both standard and vertex form, you may have noticed that the y-intercept value is equal to the value of the c constant in the equation itself. That is going to be true with every parabola/quadratic equation you encounter in those forms.
Simply look for the c constant and that is going to be your y-intercept. You can double check by using the x value of zero method.
- Math Is Fun: Quadratic Equations
- Kendall Hunt Education: Factored Form
- California State University Northridge: Parabolas
- New Paltz Central School District: Quadratic Equations in Three Forms
- ThoughtCo: How to Find the Y-Intercept of a Parabola
- University of Georgia: The Quadratic Function
- Lumen Learning: Intercepts of Quadratic Functions