1Cademy - Finding the Intercepts of $$y = 4x^2

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Finding the Intercepts of a Parabola

Example

Finding the Intercepts of $y = 4x^2 - 12x + 9$

To find the intercepts of the parabola $y = 4x^2 - 12x + 9$ :

Finding the y-intercept: Let $x = 0$ and solve for $y$ :

$y = 4(0)^2 - 12(0) + 9 = 9$

When $x = 0$ , $y = 9$ . The y-intercept is the point $(0, 9)$ .

Finding the x-intercepts: Let $y = 0$ and solve for $x$ :

$0 = 4x^2 - 12x + 9$

First, compute the discriminant to predict how many x-intercepts exist. Identify $a = 4$ , $b = -12$ , $c = 9$ :

$b^2 - 4ac = (-12)^2 - 4 \cdot 4 \cdot 9 = 144 - 144 = 0$

Because the discriminant equals zero, there is exactly one real solution, so the parabola has one x-intercept. The trinomial $4x^2 - 12x + 9$ is a perfect square trinomial that factors as:

$0 = (2x - 3)^2$

Apply the Zero Product Property:

$2x - 3 = 0$

Solve for $x$ :

$x = \frac{3}{2}$

The x-intercept is the point $\left(\frac{3}{2}, 0\right)$ .

This example illustrates the case where the discriminant equals zero, meaning the parabola just touches the x-axis at a single point rather than crossing it. Recognizing the perfect square trinomial structure allows the equation to be solved efficiently by factoring.

Updated 2026-04-21

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OpenStax Elementary Algebra

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