Lesson 3: Some Basic Graphs

New Concept

Welcome to your new toolkit of functions! This lesson introduces eight essential graphs, like the V-shaped $|x|$ and the S-curved $x^3$. Mastering their fundamental shapes is the first step to analyzing more complex functions and their transformations.

What’s next

Now, let's put these graphs into practice. You'll work through interactive examples and sketching exercises to master these eight essential shapes.

Property

$b$ is the cube root of $a$ if $b$ cubed equals $a$. In symbols, we write

Unlike square roots, which are not real for negative numbers, every real number has a real cube root. Simplifying radicals occurs at the same level as powers in the order of operations.

Examples

• To simplify $3\sqrt[3]{-8}$, we find the cube root of $-8$ which is $-2$, and then multiply by $3$. So, $3\sqrt[3]{-8} = 3(-2) = -6$.
• To evaluate $2 - \sqrt[3]{-125}$, we first find that the cube root of $-125$ is $-5$. The expression becomes $2 - (-5) = 7$.
• To simplify $\frac{10 + \sqrt[3]{-27}}{7}$, we calculate $\sqrt[3]{-27} = -3$. The expression becomes $\frac{10 + (-3)}{7} = \frac{7}{7} = 1$.

Explanation

A cube root is the inverse operation of cubing a number. Think of it as asking: 'What number, when multiplied by itself three times, gives me this value?' Unlike square roots, you can take the cube root of negative numbers.

Property

The absolute value of $x$ represents the distance from $x$ to the origin on the number line. Because distance is never negative, the absolute value is always non-negative. It is defined piecewise:

Examples

• To simplify $|3 - 8|$, first perform the operation inside the bars: $|3 - 8| = |-5|$. Then take the absolute value, which is $5$.
• In the expression $|3| - |8|$, we evaluate each absolute value first: $3 - 8$. The result is $-5$. This shows that $|a-b|$ is not always the same as $|a|-|b|$.
• To simplify $5 + 2|-4|$, we first evaluate the absolute value: $|-4| = 4$. The expression becomes $5 + 2(4) = 5 + 8 = 13$.

Explanation

Absolute value essentially makes any number positive. It measures a number's distance from zero on a number line, and distance is always a positive concept. Whether a number is positive or negative, its absolute value is its positive counterpart.

Property

The basic cubic function is given by the equation $f(x) = x^3$. Its graph has a characteristic S-shape that passes through the origin. Unlike the parabola $y=x^2$, the cubic function produces negative output values for negative input values.

Examples

• For the function $f(x) = x^3$, the point where $x=-2$ is found by calculating $f(-2) = (-2)^3 = -8$. This gives the guide point $(-2, -8)$.
• The y-intercept of $f(x) = x^3$ is at $x=0$. We have $f(0) = 0^3 = 0$. The graph passes through the origin $(0, 0)$.
• For $f(x) = x^3$, the point where $x=2$ is $f(2) = 2^3 = 8$. This gives the guide point $(2, 8)$, showing how steeply the graph rises.

Explanation

This function cubes its input value. Its S-shaped graph shows that for negative $x$, the output is negative, and for positive $x$, the output is positive. The graph grows faster than a parabola for $x > 1$.

Property

The basic reciprocal functions are $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x^2}$. An asymptote is a line that the graph of a function approaches but never touches. For these functions, the y-axis ($x=0$) is a vertical asymptote, and the x-axis ($y=0$) is a horizontal asymptote.

Examples

• For $f(x) = \frac{1}{x}$, as $x$ gets very large, like $x=1000$, $f(x)$ becomes very small, $f(1000) = 0.001$. This shows the graph approaching the horizontal asymptote $y=0$.
• For $g(x) = \frac{1}{x^2}$, as $x$ approaches $0$ from either side, like $x=0.1$ or $x=-0.1$, $g(x)$ becomes a large positive number, $g(0.1)=100$. This shows the vertical asymptote at $x=0$.
• The function $f(x) = \frac{1}{x}$ is undefined at $x=0$ because division by zero is not allowed. This is why the graph exists as two separate branches and never crosses the y-axis.

Explanation

These graphs show what happens when you divide 1 by a number. As the number gets close to zero, the result shoots towards infinity. As the number gets huge, the result shrinks towards zero, but never quite reaches it.