Intercepts of Quadratic Functions

Learning Outcomes

  • Find the [latex]y[/latex]-intercept of a quadratic function.
  • Find the real-number [latex]x[/latex]-intercepts, or roots of a quadratic function using factoring and the quadratic formula.

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the [latex]y[/latex]-intercept of a quadratic by evaluating the function at an input of zero, and we find the [latex]x[/latex]-intercepts at locations where the output is zero. Notice that the number of [latex]x[/latex]-intercepts can vary depending upon the location of the graph.

Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one –intercept, and the third parabola is of two x-intercepts.

Number of [latex]x[/latex]-intercepts of a parabola

Tip for success

We’ve built up several names for the values where a function crosses the horizontal axis:

  • horizontal intercepts or x-intercepts
  • zeros
  • roots

We tend to talk about intercepts with regard to a graph, zeros as solutions to an equation, and roots with regard to a function. Each term refers to the same value, though, for each quadratic function.

How To: Given a quadratic function [latex]f\left(x\right)[/latex], find the y– and x-intercepts.

  1. Evaluate [latex]f\left(0\right)[/latex] to find the [latex]y[/latex]-intercept.
  2. Solve the quadratic equation [latex]f\left(x\right)=0[/latex] to find the [latex]x[/latex]-intercepts.

Example: Finding the y– and x-Intercepts of a Parabola

Find the [latex]y[/latex]– and [latex]x[/latex]-intercepts of the quadratic [latex]f\left(x\right)=3{x}^{2}+5x - 2[/latex].

In the above example the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

How To: Given a quadratic function, find the x-intercepts by rewriting in standard form.

  1. Substitute [latex]a[/latex] and [latex]b[/latex] into [latex]h=-\dfrac{b}{2a}[/latex].
  2. Substitute [latex]x=h[/latex] into the general form of the quadratic function to find [latex]k[/latex].
  3. Rewrite the quadratic in standard form using [latex]h[/latex] and [latex]k[/latex].
  4. Solve for when the output of the function will be zero to find the [latex]x[/latex]intercepts.

Example: Finding the Roots of a Parabola

Find the [latex]x[/latex]-intercepts of the quadratic function [latex]f\left(x\right)=2{x}^{2}+4x - 4[/latex].

Try It

Using an online graphing calculator, plot the function [latex]g\left(x\right)=-7+{x}^{2}-6x[/latex]. You can use this tool to find the [latex]x[/latex]-and [latex]y[/latex]-intercepts by clicking on the graph. Four points will appear. List each point, and what kind of point it is, we got you started with the vertex:

  1. Vertex = [latex](3,-16)[/latex]

tip for success

When attempting to identify the roots of a quadratic function, first look to see if it can be factored when the function value is set equal to zero, and solved by applying the zero-product principle. If so, this is often the quickest method. But most quadratic functions cannot be factored. In this case, you can either write it in vertex form or use the quadratic formula.

Example: Solving a Quadratic Equation with the Quadratic Formula

Solve [latex]{x}^{2}+x+2=0[/latex].

Example: Applying the Vertex and x-Intercepts of a Parabola

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation [latex]H\left(t\right)=-16{t}^{2}+80t+40[/latex].

a. When does the ball reach the maximum height?

b. What is the maximum height of the ball?

c. When does the ball hit the ground?

Try It

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation [latex]H\left(t\right)=-16{t}^{2}+96t+112[/latex].

a. When does the rock reach the maximum height?

b. What is the maximum height of the rock?

c. When does the rock hit the ocean?