How to Graph a Parabola: Step-by-Step

How to Graph a Parabola

Graph a parabola by finding the vertex, drawing the axis of symmetry, marking the y-intercept, finding the x-intercepts, and connecting the points into a smooth U-shaped curve. A parabola is the graph of a quadratic equation y = ax² + bx + c, and its vertex sits at x = −b ÷ 2a. For y = x² − 4x + 3, the vertex is (2, −1), and the curve crosses the x-axis at 1 and 3.

The shape of every parabola comes from the squared term, which makes the curve rise on both sides of a single turning point. The Graphing Calculator plots a parabola the moment its equation is typed, and this guide finds the key points by hand so the graph can be drawn and understood without the tool.

The Parts of a Parabola

A parabola has 4 key features that fix its position and shape:

1. Vertex, the turning point where the curve changes direction, either its lowest or highest point.
2. Axis of symmetry, the vertical line through the vertex that mirrors the two halves.
3. Y-intercept, where the curve crosses the y-axis, equal to the constant c.
4. X-intercepts, where the curve crosses the x-axis, found by solving the quadratic.

These features come from the three coefficients a, b, and c. The coefficient a sets the direction and width, b shifts the vertex left or right, and c marks the y-intercept. Reading the coefficients first tells you what the parabola looks like before any point is plotted.

Step 1: Find the Direction and Width

The coefficient a decides which way a parabola opens and how wide it is. A positive a opens the parabola upward into a U shape with a lowest point, and a negative a opens it downward into an inverted U with a highest point. In y = x² − 4x + 3, the coefficient a is 1, so the parabola opens upward.

The size of a controls the width. A large absolute value, such as a = 4, produces a narrow parabola, while a small value, such as a = 0.25, produces a wide one. Reading a first sets expectations for the whole graph, so an upward-opening result with a vertex below the x-axis must cross the axis twice.

Step 2: Find the Vertex and Axis of Symmetry

The vertex x-coordinate is x = −b ÷ 2a, and substituting it back into the equation gives the y-coordinate. For y = x² − 4x + 3, the coefficient b is −4, so x = −(−4) ÷ (2 × 1) = 4 ÷ 2 = 2. Substituting x = 2 gives y = 4 − 8 + 3 = −1, so the vertex is (2, −1).

The axis of symmetry is the vertical line through the vertex, written x = 2 for this parabola. Every point on one side of this line has a mirror-image point at the same height on the other side. This symmetry means that once the vertex and one more point are known, a second point comes free by reflecting across the axis.

Step 3: Find the Intercepts

The y-intercept is the constant c, because setting x to 0 leaves y = c. For y = x² − 4x + 3, the y-intercept is 3, at the point (0, 3). This point is the fastest to plot, since it is read straight from the equation.

The x-intercepts come from setting y to 0 and solving ax² + bx + c = 0. Factoring x² − 4x + 3 gives (x − 1)(x − 3) = 0, so the x-intercepts are x = 1 and x = 3. A quadratic with no real solutions has a parabola that never crosses the x-axis, and a single repeated solution touches the axis at one point.

Worked Example: Graph y = x² − 4x + 3

Bring the steps together to graph y = x² − 4x + 3.

1. Direction: a = 1 is positive, so the parabola opens upward.
2. Vertex: x = 4 ÷ 2 = 2, and y = −1, so the vertex is (2, −1).
3. Axis of symmetry: the line x = 2.
4. Y-intercept: c = 3, at (0, 3).
5. X-intercepts: (x − 1)(x − 3) = 0, so x = 1 and x = 3.

Plot the vertex (2, −1), the y-intercept (0, 3), and the roots (1, 0) and (3, 0). Reflect the y-intercept across the axis of symmetry to get a matching point at (4, 3). Draw a smooth curve through the points to complete the parabola. The graphing calculator simulator plots the same curve and marks the vertex and roots when the equation is entered.

Graphing From Vertex Form

Vertex form writes a parabola as y = a(x − h)² + k, where (h, k) is the vertex. This form shows the vertex directly, with no calculation. The equation y = (x − 2)² − 1 is the vertex form of y = x² − 4x + 3, so the vertex (2, −1) is read straight from the h and k values.

Converting standard form to vertex form uses completing the square, but for graphing, vertex form saves the vertex step entirely. Plot the vertex from h and k, use a to set the direction and width, then find one more point to fix the curve. The sibling guide on how to graph a linear equation covers the straight-line case that vertex form builds on.

Frequently Asked Questions

How do you graph a parabola step by step?

Find the vertex, draw the axis of symmetry, mark the y-intercept, find the x-intercepts if they exist, then draw a smooth U-shaped curve through the points. For y = ax² + bx + c, the vertex x-value is −b divided by 2a. Plot the vertex, use the axis of symmetry to mirror points, and connect them into a parabola.

How do you find the vertex of a parabola?

Find the vertex x-coordinate with the formula x equals −b divided by 2a, then substitute that x back into the equation to get the y-coordinate. For y = x² − 4x + 3, x equals 4 divided by 2, which is 2, and y equals 4 minus 8 plus 3, which is −1, so the vertex is (2, −1). The vertex is the lowest or highest point of the parabola.

Which way does a parabola open?

A parabola opens upward when the coefficient a is positive and downward when a is negative. In y = ax² + bx + c, the sign of a alone decides the direction. A positive a gives a U shape with a lowest point at the vertex, and a negative a gives an inverted U with a highest point at the vertex.

How do you find the x-intercepts of a parabola?

Set y to 0 and solve the quadratic ax² + bx + c = 0 by factoring or the quadratic formula. The solutions are the x-intercepts, where the parabola crosses the x-axis. For y = x² − 4x + 3, factoring gives (x − 1)(x − 3) = 0, so the x-intercepts are at x = 1 and x = 3. A parabola can have two, one, or no x-intercepts.

What is the axis of symmetry of a parabola?

The axis of symmetry is the vertical line that splits the parabola into two mirror-image halves, and it passes through the vertex. Its equation is x equals −b divided by 2a, the same value as the vertex x-coordinate. For y = x² − 4x + 3, the axis of symmetry is the line x = 2, so points equal distances left and right of it share the same y-value.