Lesson 3: Analyzing Functions Graphically

Property

Domain: The set of all the $x$-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of $x$ that have a corresponding value on the graph.
Range: The set of all the $y$-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of $y$ that have a corresponding value on the graph.

Examples

• If a graph extends horizontally from $x=-5$ to $x=5$, its domain is $[-5, 5]$. If it extends vertically from $y=0$ to $y=10$, its range is $[0, 10]$.
• To find $f(2)$ from a graph, locate $x=2$ on the horizontal axis, move up to the graph, and read the y-value. If the point is $(2, 4)$, then $f(2)=4$.
• If a graph touches the x-axis at $x=-3$, the x-intercept is $(-3, 0)$. If it crosses the y-axis at $y=6$, the y-intercept is $(0, 6)$.

Explanation

A graph provides a visual summary of a function. Its horizontal spread is the domain, and its vertical reach is the range. You can also find specific outputs ($y$-values) for given inputs ($x$-values).

Property

A function has a maximum value when there is a $y$-value that is greater than or equal to all other $y$-values in the function''s range. A function has a minimum value when there is a $y$-value that is less than or equal to all other $y$-values in its range. These values are also known as global or absolute maximums and minimums.

Examples

• The function $f(x) = -x^2 + 4$ has a maximum value of $4$. It has no minimum value.
• The function $g(x) = |x| - 2$ has a minimum value of $-2$. It has no maximum value.
• The function $h(x) = \sin(x)$ has a maximum value of $1$ and a minimum value of $-1$.

Explanation

The maximum and minimum are the "highest" and "lowest" points on the entire graph of a function. The maximum value is the largest output ($y$-value) the function can produce, while the minimum value is the smallest output. Not all functions have a maximum or minimum value; for example, a line like $y=x$ continues infinitely in both positive and negative $y$ directions.

Property

The axis of symmetry is a vertical line that divides a graph into two congruent, mirror-image halves. The equation of this vertical line is given by $x = h$, where $h$ is a constant. For any point on the graph, there is a corresponding point on the opposite side of this line that is equidistant from it.

Examples

• If a parabola has an axis of symmetry at $x = 2$ and a point at $(0, 5)$, there must be a corresponding mirror-image point at $(4, 5)$.
• An absolute value function with a vertex at $(-3, 1)$ has an axis of symmetry with the equation $x = -3$.
• If a graph has an axis of symmetry at $x=0$ (the y-axis), then for any point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph.

Explanation

The axis of symmetry is a fundamental property of certain functions, most notably quadratic and absolute value functions. It is a vertical line that passes through the vertex of the graph. Understanding the axis of symmetry allows you to predict the location of points on the graph, as every point (except those on the axis itself) has a matching counterpart on the other side.