How do you graph a quadratic equation?

Definition

A quadratic equation is any equation of the form
$$ y = ax^2 + bx + c $$
where $a \neq 0$, and $a$, $b$, $c$ are real numbers. The graph of a quadratic equation is a parabola that opens upward if $a > 0$ and downward if $a < 0$.

Steps to Graph a Quadratic Equation

1. Find the vertex: The vertex $(h, k)$ is given by

$$ h = -\frac{b}{2a}, \quad k = a h^2 + b h + c $$

2. Find the axis of symmetry: This is the vertical line $x = h$.
3. Find the y-intercept: Set $x = 0$ to get $y = c$.
4. Find the x-intercepts (if any): Solve $ax^2 + bx + c = 0$ using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

5. Plot the points and sketch the parabola.

Worked Example

Graph $y = x^2 - 4x + 3$.

Step 1: Vertex

• $a = 1$, $b = -4$, $c = 3$
• $h = -\frac{-4}{2 \cdot 1} = 2$
• $k = (2)^2 - 4 \cdot 2 + 3 = 4 - 8 + 3 = -1$
• Vertex: $(2, -1)$

Step 2: Axis of symmetry

• $x = 2$

Step 3: y-intercept

• $y = 0^2 - 4 \cdot 0 + 3 = 3$
• Point: $(0, 3)$

Step 4: x-intercepts

• Solve $x^2 - 4x + 3 = 0$
• $x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 3}}{2} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2}$
• $x = 3$ and $x = 1$
• Points: $(1, 0)$ and $(3, 0)$

Step 5: Sketch

• Plot the vertex, axis of symmetry, y-intercept, and x-intercepts. Draw a symmetric parabola opening upward.

Takeaways

• The vertex and axis of symmetry are key to graphing quadratics.
• Use intercepts to plot accurate points.
• The sign of $a$ determines if the parabola opens up or down.