Math Core

Throw a ball, aim a water fountain, or trace the cable of a suspension bridge, and you get the same U-shaped curve. That curve is the graph of a quadratic function, and reading its key features (where it turns, where it crosses the axes, how wide it is) is the first step to solving every quadratic equation in this unit.

What makes a function quadratic

In the linear functions unit, the highest power of $x$ was $1$ . A quadratic function has an $x^2$ term, and nothing higher.

Definition

Quadratic function

A quadratic function can be written in standard form

f(x) = ax^2 + bx + c,

where $a$ , $b$ and $c$ are real numbers and $a \ne 0$ . Its graph is a U-shaped curve called a parabola.

The condition $a \ne 0$ matters: if $a = 0$ , the $x^2$ term disappears and you are left with the line $y = bx + c$ . The other coefficients may be zero. For example, $y = x^2$ , $y = 3x^2 - 7$ and $y = -x^2 + 4x$ are all quadratic.

The parent function $y = x^2$

The simplest quadratic is $y = x^2$ . Make a table of values:

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y = x^2$	$9$	$4$	$1$	$0$	$1$	$4$	$9$

Notice that the outputs repeat in pairs: $x = 2$ and $x = -2$ both give $4$ , because squaring erases the sign. That repetition is why every parabola is symmetric.

The parent function y = x². Its lowest point, the vertex, is at the origin, and the two halves mirror each other across the y-axis.Open in grapher →

The features of a parabola

Every parabola has the same set of landmarks. Here they are on the graph of $y = x^2 - 2x - 3$ .

y = x² - 2x - 3 has vertex (1, -4), axis of symmetry x = 1 (dashed), x-intercepts -1 and 3, and y-intercept -3.Open in grapher →

The vertex is the turning point, here $(1, -4)$ . If the parabola opens up, the vertex is the minimum point; if it opens down, the vertex is the maximum point.
The axis of symmetry is the vertical line through the vertex, here $x = 1$ . Fold the graph along it and the two halves match.
The y-intercept is where the graph crosses the $y$ -axis. Setting $x = 0$ in $ax^2 + bx + c$ leaves $c$ , so the y-intercept is always $(0, c)$ . Here it is $(0, -3)$ .
The x-intercepts, also called zeros or roots, are where $y = 0$ . Here they are $x = -1$ and $x = 3$ . A parabola can have two, one or no x-intercepts.

The domain of every quadratic function is all real numbers. The range depends on the vertex: this parabola opens up from a lowest value of $-4$ , so its range is $y \ge -4$ .

How $a$ shapes the parabola

The leading coefficient $a$ controls the direction and the width.

If $a > 0$ , the parabola opens up (a minimum). If $a < 0$ , it opens down (a maximum).
If $|a| > 1$ , the parabola is narrower than $y = x^2$ . If $0 < |a| < 1$ , it is wider.

A larger |a| squeezes the parabola; a smaller |a| stretches it. A negative a flips it upside down.Open in grapher →

Finding the vertex from standard form

You can find the vertex without graphing. Start with the y-intercept $(0, c)$ . Which other point on the parabola has the same height $c$ ? Solve $ax^2 + bx + c = c$ :

ax^2 + bx = 0 \quad\Longrightarrow\quad x(ax + b) = 0 \quad\Longrightarrow\quad x = 0 \ \text{ or } \ x = -\dfrac{b}{a}.

These two points are mirror images, so the axis of symmetry is exactly halfway between them, at $x = -\dfrac{b}{2a}$ .

Vertex of y = ax² + bx + c

The axis of symmetry is the line

x = -\frac{b}{2a}.

The vertex lies on this line. To find its $y$ -coordinate, substitute this $x$ -value back into the function.

Worked example: Vertex, axis and intercepts

Find the vertex, axis of symmetry and y-intercept of $y = x^2 - 6x + 5$ , then sketch the graph.

Here $a = 1$ and $b = -6$ , so

x = -\frac{-6}{2(1)} = 3.

Substitute $x = 3$ : $y = 9 - 18 + 5 = -4$ . The vertex is $(3, -4)$ , and since $a > 0$ it is a minimum. The axis of symmetry is $x = 3$ .

The y-intercept is $(0, 5)$ . Its mirror image across $x = 3$ is $(6, 5)$ , three units on the other side. Checking a couple more points, $x = 1$ and $x = 5$ both give $y = 0$ , so those are the x-intercepts.

y = x² - 6x + 5: vertex (3, -4), axis x = 3, y-intercept (0, 5) and its mirror point (6, 5).Open in grapher →

Common mistake

The formula is $x = -\dfrac{b}{2a}$ , with a minus sign in front. When $b$ is already negative, the two negatives make a positive: for $y = x^2 - 6x + 5$ , the axis is $x = +3$ , not $-3$ . Also remember that $-\dfrac{b}{2a}$ gives only the $x$ -coordinate of the vertex; you still have to substitute to get $y$ .

Worked example: A parabola that opens down

Find the vertex and range of $y = -2x^2 - 8x - 3$ .

Here $a = -2$ and $b = -8$ :

x = -\frac{-8}{2(-2)} = \frac{8}{-4} = -2.

Then $y = -2(-2)^2 - 8(-2) - 3 = -8 + 16 - 3 = 5$ . The vertex is $(-2, 5)$ .

Since $a < 0$ , the parabola opens down, so $5$ is the maximum value. The range is $y \le 5$ .

Graphing step by step

To sketch any parabola $y = ax^2 + bx + c$ :

Check the sign of $a$ to see which way it opens.
Find the axis of symmetry $x = -\dfrac{b}{2a}$ and the vertex.
Plot the y-intercept $(0, c)$ and reflect it across the axis.
Plot one or two more points if needed, reflect them too, and draw a smooth U through all of them.

Tip

Once you know the vertex, you know the range for free: $y \ge k$ if the parabola opens up, $y \le k$ if it opens down, where $k$ is the $y$ -coordinate of the vertex.

Worked example: The highest point of a throw

A ball is tossed upward. Its height in feet after $t$ seconds is $h(t) = -16t^2 + 48t + 4$ . When does it reach its highest point, and how high is that?

The maximum is at the vertex. With $a = -16$ and $b = 48$ :

t = -\frac{48}{2(-16)} = \frac{48}{32} = 1.5.

Then $h(1.5) = -16(2.25) + 48(1.5) + 4 = -36 + 72 + 4 = 40$ . The ball reaches a maximum height of $40$ feet after $1.5$ seconds. The y-intercept $h(0) = 4$ tells you it was released $4$ feet above the ground.

Height (y, in feet) against time (x, in seconds). The vertex (1.5, 40) is the top of the throw.Open in grapher →

Practice

Practice 1

Which parabola opens downward?

Practice 2

The axis of symmetry of $y = x^2 + 8x - 1$ is the line $x = ?$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Find the vertex of $y = x^2 - 4x + 7$ .

Enter a point like (2, -3)

Practice 4

Which parabola is the narrowest?

Practice 5

The graph of $y = -x^2 + 2x + 3$ is shown. What are its zeros?

y = -x^2 + 2x + 3(1, 4)Open in grapher →

Separate answers with commas, e.g. 2, -5

Practice 6

What is the maximum value of $f(x) = -3x^2 + 12x - 5$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Find the range of $y = 2x^2 + 4x - 1$ . Write it as an inequality in $y$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Practice 8

A ball's height in feet after $t$ seconds is $h(t) = -16t^2 + 64t + 5$ . What is its maximum height, in feet?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

What makes a function quadratic

The parent function y=x2y = x^2y=x2

The features of a parabola

How aaa shapes the parabola

Finding the vertex from standard form

Graphing step by step

Practice

The parent function $y = x^2$

How $a$ shapes the parabola