Analyzing Graphs

Math 1: Analyzing Graphs

by Kaitlin Murphy

​Characteristics of Graphs

​All graphs are different!

​So, we have a lot of ways to describe what they look like and mean.

​

​This lesson will go over some of the vocab used to describe characteristics of graphs.

​y-intercept

The y-intercept is the place where the graph of the function crosses the y-axis.

​

​The function to the right has a

y-intercept of -6.

​

​Sometimes we use b as the variable for the y-intercept.

Multiple Choice

What is the y-intercept shown on the graph?

• (0, -2)

• (3, 0)

• (0, 3)

• None of these.

Multiple Choice

• b=2

• m=3

• b=-3

• m=-2

​x-intercept(s)

An x-intercept is any place at which the graph of a function crosses the x-axis. It is possible for functions to have more than one x-intercepts (or zero!).

​

​The function to the right has three

x-intercepts at -3, -2, and 1.

​

​X-intercepts can also be called roots, solutions, or zeroes.

Multiple Choice

What are all the x-intercepts?

• (-2.55, 0), (2.55, 0)

• (-3,0), (-2,0), (2,0), (3,0)

• (0, 36)

• (-∞,∞)

Multiple Choice

Find the zeros.

• x = 0 and 4

• x = 0 and -4

• x = 0

• x = -4

Multiple Select

What are the zeros for the given polynomial? (Check all that apply)

HInt: where does the graph cross the X-axis

• -4

• -3

• -2

• -1

• 2

​Positive/Negative Intervals

It can be interesting to tell when a function is positive (above the x-axis) vs. when it is negative (below the x-axis). Remember to read the graph from left to right!

​

​The function to the right is negative on the intervals (-infinity, -3) and (-2,1).

​It is positive from (-3,-2) and (1, infinity).

Multiple Choice

Write the interval that represents the negative portion of the function?

• (-2,1)

• (-0.5,-3.5)

• $$\left(-\infty,-2\right);\ \left(1,\infty\right)$$

• (-3,0)

Multiple Choice

Where is the function positive?

• (-∞,5]

• $$\left(0,\ \infty\right)$$

• (-∞,2]

• (-∞,0)

Multiple Choice

Where is the function negative?

• (-∞,1)

• (-∞,∞)

• (1,∞) and (-∞,1)

• (-1,∞)

​Increasing/Decreasing Intervals

Besides positive and negative, it can be interesting to tell when a function is increasing vs. decreasing.

​

​The function to the right is increasing from (-infinity, -2.5ish) and (0, infinity).

​It is decreasing (sloping downward) from (-2.5, 0).

Multiple Choice

What is the increasing interval for the function below?

• $$\left(-\infty,\infty\right)$$

• $$\left(-\infty,2\right]$$

• $$\left(-\infty,6\right]$$

• $$\left(-6,6\right)$$

Multiple Choice

What is/are the increasing interval(s) for the function shown?

• $$\left(-3,\ 1\right)$$

• $$\left(4,\ 8\right)$$

• $$\left(-\infty,1\right)\ and\ \left(-3,\ 1\right)$$

Multiple Choice

What is the decreasing interval on the function shown?

• $$\left(-\infty,\ 1\right)$$

• $$\left(-\infty,\ 2\right)$$

• $$\left(2,\ \infty\right)$$

• $$\left(1,\ \infty\right)$$

Multiple Choice

How many increasing and decreasing intervals are there in this graph?

• 2 increasing, 2 decreasing

• 1 increasing, 2 decreasing

• 2 increasing, 1 decreasing

• 1 increasing, 1 decreasing

Multiple Choice

Over what interval is this function constant?

• -5 < x < ∞

• -3 < x < 4

• 4 < x < ∞

• -5

maximum and minimum

Often we want to know when the graph reaches a maximum or minimum value.

​

​​The graph to the right has a local max at -2.5 with a value of about 1.

​​It also has a local minimum at 0 with a value of about -6.

​

​A global max/min would be the highest or lowest point of the entire graph. This graph does not have those since it keeps going down forever to the left and up forever to the right.

Multiple Choice

Does this graph have a minimum or maximum? What point?

• minimum at (-3, -5)

• minimum at (3, 5)

• maximum at (-3, -5)

• maximum at (3, 5)

Multiple Choice

Does this graph have a minimum or maximum? What point?

• Minimum at (0, 0)

• Minimum at (0, 2)

• Maximum at (0, 0)

• Maximum at (0, 2)

Multiple Choice

Which ordered pair identifies the *absolute minimum* point on this function?

• (-1, 0)

• (0, -3)

• (1, -4)

• (4, 0)

Multiple Choice

Which coordinate would qualify as the *absolute maximum* of this function?

• ∞

• (0, 4)

• 3

• (4, 0)

Multiple Choice

What is the minimum value (y-values only)?

• 0

• -5

• -3

• -1

Multiple Choice

What is the maximum value (y-value only)?

• 2

• 5

• -1

• 0