Precalculus Functions and Their Graphs Syllabus

What is the algebraic test for a function $f$ to be even?

$f$ is even if $f(-x) = f(x)$ for all $x$ in the domain; its graph is symmetric about the $y$-axis.

What is the algebraic test for a function $f$ to be odd?

$f$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain; its graph is symmetric about the origin.

Classify $f(x) = x^{3} - 4x$ as even, odd, or neither.

$f(-x) = (-x)^{3} - 4(-x) = -x^{3} + 4x = -(x^{3}-4x) = -f(x)$, so $f$ is odd.

A graph is symmetric about the $x$-axis. Can it represent a function? Why or why not?

Generally no. $x$-axis symmetry means points $(x, y)$ and $(x, -y)$ both lie on the graph, so most inputs have two outputs, violating the vertical line test.

Write the formula for the average rate of change of $f$ from $x = a$ to $x = b$.

$$\frac{f(b) - f(a)}{b - a}$$

Find the average rate of change of $f(x) = x^{2}$ from $x = 1$ to $x = 4$.

$$\frac{f(4)-f(1)}{4-1} = \frac{16 - 1}{3} = \frac{15}{3} = 5$$

What does the average rate of change of $f$ over $[a,b]$ equal geometrically?

The slope of the secant line joining the points $(a, f(a))$ and $(b, f(b))$ on the graph.

What is the general form of a linear function and what does each parameter represent?

$f(x) = mx + b$, where $m$ is the slope (constant rate of change) and $b$ is the $y$-intercept.

What is a constant function, and what does its graph look like?

A function of the form $f(x) = c$ for a fixed real number $c$. Its graph is a horizontal line, and its slope is $0$.

What is the defining property of the average/instantaneous rate of change of a linear function $f(x)=mx+b$?

It is constant everywhere and equals the slope $m$, regardless of the interval chosen.

Describe the shape and domain/range of the identity function $f(x) = x$.

A straight line through the origin with slope $1$; domain and range are both $(-\infty, \infty)$. It is odd and increasing everywhere.

Describe the graph of the power function $f(x) = x^{2}$ (squaring function).

A parabola opening upward with vertex at the origin; domain $(-\infty,\infty)$, range $[0,\infty)$; even function, symmetric about the $y$-axis.

Describe the graph of the cubing function $f(x) = x^{3}$.

An S-shaped curve through the origin; domain and range both $(-\infty,\infty)$; odd function, symmetric about the origin, increasing everywhere.

State the domain and range of the square root function $f(x) = \sqrt{x}$.

Domain $[0, \infty)$ and range $[0, \infty)$; the graph starts at the origin and increases.

State the domain and range of the cube root function $f(x) = \sqrt[3]{x}$.

Both domain and range are $(-\infty, \infty)$; it is an odd function symmetric about the origin.

Give the piecewise definition of the absolute value function $f(x) = |x|$.

$$|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$$

Describe the graph, domain, and range of $f(x) = |x|$.

A V-shape with vertex at the origin; domain $(-\infty,\infty)$, range $[0,\infty)$; even function, decreasing on $(-\infty,0]$ and increasing on $[0,\infty)$.

Define the greatest integer (floor) function $f(x) = \lfloor x \rfloor$.

$\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example, $\lfloor 2.7 \rfloor = 2$ and $\lfloor -1.3 \rfloor = -2$.

What type of graph does the greatest integer function produce, and what is its range?

A step function consisting of horizontal segments (steps) that jump at each integer; its range is the set of all integers $\mathbb{Z}$.

Write the standard (vertex) form of a quadratic function and identify the vertex.

$f(x) = a(x-h)^{2} + k$ with $a \neq 0$; the vertex is $(h, k)$ and the axis of symmetry is $x = h$.

For $f(x) = ax^{2} + bx + c$, give the $x$-coordinate of the vertex.

$$x = -\frac{b}{2a}$$, and the $y$-coordinate is $f\!\left(-\frac{b}{2a}\right)$.

How does the sign of $a$ affect the parabola $f(x)=ax^{2}+bx+c$?

If $a > 0$ the parabola opens upward and the vertex is a minimum; if $a < 0$ it opens downward and the vertex is a maximum.

How does the graph of $y = f(x) + c$ (with $c>0$) compare to $y = f(x)$?

It is a vertical shift upward by $c$ units. $y = f(x) - c$ shifts the graph downward by $c$ units.

How does the graph of $y = f(x - c)$ (with $c>0$) compare to $y = f(x)$?

It is a horizontal shift to the right by $c$ units. $y = f(x + c)$ shifts the graph left by $c$ units.

Describe the transformation from $y = f(x)$ to $y = -f(x)$.

A reflection across the $x$-axis; each output is negated so the graph flips vertically.

Describe the transformation from $y = f(x)$ to $y = f(-x)$.

A reflection across the $y$-axis; each input is negated so the graph flips horizontally.

For $y = a\,f(x)$ with $a > 1$ versus $0 < a < 1$, describe the transformation.

If $a > 1$ it is a vertical stretch by factor $a$; if $0 < a < 1$ it is a vertical compression (shrink) toward the $x$-axis.

For $y = f(bx)$ with $b > 1$ versus $0 < b < 1$, describe the horizontal transformation.

If $b > 1$ it is a horizontal compression by factor $\frac{1}{b}$; if $0 < b < 1$ it is a horizontal stretch by factor $\frac{1}{b}$.

In what order should transformations generally be applied to graph $y = a\,f(b(x-h)) + k$?

Horizontal shifts/stretches/reflections first (inside the function), then vertical stretches/reflections, and finally vertical shifts. A common order is: horizontal shift, stretch/compress, reflect, then vertical shift.

Given $f$ and $g$, write the definitions of $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\left(\frac{f}{g}\right)(x)$.

$(f+g)(x)=f(x)+g(x)$, $(f-g)(x)=f(x)-g(x)$, $(fg)(x)=f(x)\,g(x)$, and $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$ where $g(x)\neq 0$.

What is the domain of an arithmetic combination such as $(f+g)$ or $(fg)$, and how does the quotient $\frac{f}{g}$ differ?

The domain is the intersection of the domains of $f$ and $g$. For $\frac{f}{g}$ you must additionally exclude all $x$ where $g(x) = 0$.