The Quadratic Function and Its Graph, foundations for calculus lesson

Water from a garden hose, a ball tossed to a friend and the cable of a suspension bridge all trace curves shaped much like a parabola. A quadratic function is the simplest formula that bends, and it is the first curve calculus uses to show what a minimum, a maximum and a tangent line look like. The goal here is to sketch one from facts you can justify with algebra, not from a handful of plotted points and a guess.

Key terms

Quadratic function

Definition A quadratic function is a function of the form

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

where $a$ , $b$ and $c$ are real constants with $a \ne 0$ . Its domain is all real numbers. If $a = 0$ the function is linear, not quadratic.

In plain words A quadratic has an $x^2$ term that really is there, and possibly an $x$ term and a constant.

Example The function $f(x) = x^2 - 4x - 5$ is quadratic with $a = 1$ , $b = -4$ and $c = -5$ . The function $g(x) = 0x^2 + 3x + 1 = 3x + 1$ is not, because its $x^2$ coefficient is zero.

Equivalent forms

Definition A quadratic function can be written in three common forms that agree for every real $x$ . The standard form $f(x) = ax^2 + bx + c$ shows the $y$ intercept $c$ . The vertex form $f(x) = a(x - h)^2 + k$ shows the vertex $(h, k)$ . Completing the square always produces it, with

h = -\frac{b}{2a}, \qquad k = f(h) = c - \frac{b^2}{4a}.

The factored form $f(x) = a(x - r_1)(x - r_2)$ shows the zeros $r_1$ and $r_2$ . A factored form with real $r_1$ and $r_2$ exists exactly when $f$ has at least one real zero, that is, when $b^2 - 4ac \ge 0$ . If $b^2 - 4ac = 0$ the two factors are equal, $r_1 = r_2$ , and if $b^2 - 4ac < 0$ there is no factored form with real numbers.

The leading coefficient $a$ is the same in all three forms.

In plain words These are the same function wearing different clothes. Pick the form that shows the feature you need right now.

Example For $f(x) = x^2 - 4x - 5$ , completing the square and factoring give

\begin{aligned} x^2 - 4x - 5 &= (x - 2)^2 - 9 \\[0.4em] &= (x - 5)(x + 1). \end{aligned}

Expanding $(x - 2)^2 - 9$ gives $x^2 - 4x + 4 - 9$ , and expanding $(x - 5)(x + 1)$ gives $x^2 - 4x - 5$ , so all three forms match.

Parabola

Definition The graph of a quadratic function, the set of points $(x, ax^2 + bx + c)$ for real $x$ , is called a parabola. Every such graph is obtained from the graph of $y = x^2$ by a vertical stretch by the factor $\lvert a \rvert$ , a reflection across the $x$ axis when $a < 0$ , and a shift that moves the origin to the vertex $(h, k)$ .

In plain words Every quadratic graph is the basic $y = x^2$ curve, made narrower or wider, possibly flipped, and slid into place.

Example The graph of $y = 2(x - 1)^2 + 3$ is the curve $y = x^2$ stretched vertically by $2$ and moved $1$ unit right and $3$ units up. Its lowest point is $(1, 3)$ . A table of values such as $x = 0, 1, 2$ giving $y = 5, 3, 5$ is a good check on a sketch, but a table alone cannot prove where the lowest point is. The vertex form proves it.

Opening direction

Definition For every real $x$ the square satisfies $(x - h)^2 \ge 0$ , with equality exactly when $x = h$ . Therefore in $f(x) = a(x - h)^2 + k$ the parabola opens upward when $a > 0$ , so that $f(x) \ge k$ for every $x$ , and opens downward when $a < 0$ , so that $f(x) \le k$ for every $x$ .

In plain words The sign of the $x^2$ coefficient decides everything about direction. Positive opens up like a cup, negative opens down like a cap.

Example The function $g(x) = -x^2 + 2x + 3$ has $a = -1 < 0$ , so its graph opens downward. In vertex form

g(x) = -(x - 1)^2 + 4,

and since $-(x - 1)^2 \le 0$ , every output satisfies $g(x) \le 4$ .

Vertex

Definition The vertex of the parabola $y = a(x - h)^2 + k$ is the point $(h, k)$ . When $a > 0$ the value $k$ is the minimum of the function $f(x) = a(x - h)^2 + k$ , and when $a < 0$ it is the maximum. In both cases it is attained only at $x = h$ , because $(x - h)^2 = 0$ only there.

In plain words The vertex is the turning point, the bottom of an upward parabola or the top of a downward one.

Example From $f(x) = (x - 2)^2 - 9$ the vertex is $(2, -9)$ . Since $a = 1 > 0$ , the minimum value of $f$ is $-9$ , reached only at $x = 2$ . The formula $h = -\frac{b}{2a}$ agrees, giving $-\frac{-4}{2} = 2$ .

Axis of symmetry

Definition The axis of symmetry of the parabola $y = a(x - h)^2 + k$ is the vertical line $x = h$ . For every real $t$ ,

f(h + t) = a t^2 + k = f(h - t),

so points at equal distances to the left and right of the axis have equal heights.

In plain words Fold the graph along the vertical line through the vertex and the two halves land on top of each other.

Example For $f(x) = (x - 2)^2 - 9$ the axis is $x = 2$ . The inputs $0$ and $4$ are both $2$ units from the axis, and indeed $f(0) = -5$ and $f(4) = -5$ . Knowing the $y$ intercept $(0, -5)$ therefore gives the mirror point $(4, -5)$ for free.

Intercepts

Definition The $y$ intercept of $f(x) = ax^2 + bx + c$ is the point $(0, c)$ , and every quadratic has exactly one. The $x$ intercepts are the points $(r, 0)$ where $r$ is a real solution of $ax^2 + bx + c = 0$ . With the discriminant $\Delta = b^2 - 4ac$ , there are two $x$ intercepts when $\Delta > 0$ , one (the vertex) when $\Delta = 0$ , and none when $\Delta < 0$ .

In plain words The graph always crosses the $y$ axis once. It meets the $x$ axis only where the equation $f(x) = 0$ has real solutions, which can be at two points, at one point where it just touches, or never.

Example For $f(x) = x^2 - 4x - 5 = (x - 5)(x + 1)$ the $y$ intercept is $(0, -5)$ and the $x$ intercepts are $(-1, 0)$ and $(5, 0)$ . They sit symmetrically about the axis $x = 2$ , each $3$ units away. For $p(x) = x^2 + 2x + 5$ the discriminant is $4 - 20 = -16 < 0$ , so its graph never meets the $x$ axis, even though its $y$ intercept $(0, 5)$ exists.

Range

Definition A quadratic function $f(x) = a(x - h)^2 + k$ on all real numbers has range

\begin{cases} [k, \infty) & \text{if } a > 0, \\ (-\infty, k] & \text{if } a < 0. \end{cases}

The bound comes from $(x - h)^2 \ge 0$ . Every value beyond the bound is attained, since for $y$ with $(y - k)/a \ge 0$ the input

x = h + \sqrt{\frac{y - k}{a}}

satisfies $f(x) = y$ .

In plain words An upward parabola reaches every height from its vertex up, and a downward one reaches every height from its vertex down.

Example Take the downward parabola $g(x) = -(x - 1)^2 + 4 = -(x - 3)(x + 1)$ . Its outputs are at most $4$ . For any $y \le 4$ the input $x = 1 + \sqrt{4 - y}$ gives $g(x) = -(4 - y) + 4 = y$ , so the range is $(-\infty, 4]$ . The full sketch now follows from justified features, the vertex $(1, 4)$ at the top, the axis $x = 1$ , the $y$ intercept $(0, 3)$ with its mirror point $(2, 3)$ , and the $x$ intercepts $(-1, 0)$ and $(3, 0)$ .

Check yourself

Write $f(x) = 2x^2 + 8x + 6$ in vertex form and factored form. Give the vertex, opening direction, intercepts and range.

Show the answer

Completing the square and factoring give

\begin{aligned} 2x^2 + 8x + 6 &= 2(x + 2)^2 - 2 \\[0.4em] &= 2(x + 1)(x + 3). \end{aligned}

The vertex is $(-2, -2)$ and the parabola opens upward because $a = 2 > 0$ . The $y$ intercept is $(0, 6)$ , the $x$ intercepts are $(-3, 0)$ and $(-1, 0)$ , and the range is $[-2, \infty)$ .

Does the graph of $h(x) = -x^2 + 6x - 10$ cross the $x$ axis? Find its range.

Show the answer

In vertex form $h(x) = -(x - 3)^2 - 1$ . The parabola opens downward with maximum value $-1$ at $x = 3$ , so every output is negative and the graph never meets the $x$ axis. The discriminant agrees, since $36 - 40 = -4 < 0$ . The range is $(-\infty, -1]$ .

A parabola has vertex $(1, -2)$ and passes through $(3, 6)$ . Find its quadratic function.

Show the answer

Write $f(x) = a(x - 1)^2 - 2$ . The point $(3, 6)$ gives $4a - 2 = 6$ , so $a = 2$ and $f(x) = 2(x - 1)^2 - 2$ , which expands to $2x^2 - 4x$ .

The Quadratic Function and Its Graph

Key terms

Quadratic function

Equivalent forms

Parabola

Opening direction

Vertex

Axis of symmetry

Intercepts

Range

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus