Intercepts of Quadratic Functions

Learning Objectives

  • Find the y-intercept of a quadratic function
  • Find the real-number x-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 y-intercept of a quadratic by evaluating the function at an input of zero, and we find the x-intercepts at locations where the output is zero. Notice that the number of x-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 x-intercepts of a parabola

Mathematicians also define x-intercepts as roots of the 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 y-intercept.
  2. Solve the quadratic equation [latex]f\left(x\right)=0[/latex] to find the x-intercepts.

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

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

In Example: Finding the y– and x-Intercepts of a Parabola, 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 a and b into [latex]h=-\frac{b}{2a}[/latex].
  2. Substitute xh into the general form of the quadratic function to find k.
  3. Rewrite the quadratic in standard form using h and k.
  4. Solve for when the output of the function will be zero to find the x-intercepts.

Example: Finding the Roots of a Parabola

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

Try It

The function [latex]g\left(x\right)=13+{x}^{2}-6x[/latex] is graphed below. You can use Desmos to find the x-and y-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]

https://www.desmos.com/calculator/ilnmrc6noz

 

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?