April 19, 2026
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How to Graph Using the Formula

Manny04 mins
How to Graph Using the Formula How to Graph Using the Formula

The quadratic formula is useful for solving equations. But it can also help you graph a parabola. By using the formula step by step, you can find key points that help shape the curve.

In this guide, you’ll learn how to graph using the formula so you can plot any quadratic equation correctly.

Step 1: Write the Equation in Standard Form

To begin, make sure your equation is written as:

ax² + bx + c = 0

This form helps you easily identify the values of a, b, and c.

Example:
y = x² – 4x + 3
Here, a = 1, b = –4, c = 3

You will use these values in the formula.

Step 2: Use the Quadratic Formula to Find the x-Intercepts

The x-intercepts are where the graph crosses the x-axis. These are also called the roots or solutions of the equation.

To find them, use the quadratic formula:

x = (–b ± √(b² – 4ac)) / 2a

Using the example:
a = 1, b = –4, c = 3
x = (4 ± √(–4)² – 4(1)(3)) / 2(1)
x = (4 ± √(16 – 12)) / 2
x = (4 ± √4) / 2
x = (4 ± 2) / 2

Now solve both values:
x = (4 + 2)/2 = 6/2 = 3
x = (4 – 2)/2 = 2/2 = 1

So, the graph crosses the x-axis at x = 1 and x = 3.
Plot the points (1, 0) and (3, 0).

How to Graph Using the Formula
How to Graph Using the Formula

Step 3: Find the Axis of Symmetry

The axis of symmetry is a vertical line that goes through the middle of the parabola. You can find it using:

x = –b / 2a

In our example:
x = –(–4) / (2 × 1) = 4 / 2 = 2

So, the axis of symmetry is x = 2. This is the center of the graph.

Step 4: Find the Vertex

The vertex is the highest or lowest point on the graph. To find it, use the x-value from the axis of symmetry and plug it into the original equation.

Use x = 2:
y = (2)² – 4(2) + 3 = 4 – 8 + 3 = –1

So, the vertex is (2, –1).
Plot this point. It is the lowest point because the parabola opens upward.

Step 5: Plot the Y-Intercept

The y-intercept is where the graph crosses the y-axis. You find this by setting x = 0 in the equation.

y = (0)² – 4(0) + 3 = 3

So, the y-intercept is (0, 3).
Plot this point as well.

Step 6: Draw the Parabola

Now that you have:

  • The vertex at (2, –1)

  • The x-intercepts at (1, 0) and (3, 0)

  • The y-intercept at (0, 3)

  • The axis of symmetry at x = 2

Use these points to sketch a smooth U-shaped curve. Make sure the curve is symmetric across the line x = 2.

Final Thoughts

Now you know how to graph using the formula. The quadratic formula gives you the x-intercepts. From there, you find the axis of symmetry, the vertex, and the y-intercept. These points help you draw the complete graph of a parabola.

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