Objectives
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Use the graph of a function to find the domain and range in interval notation, the \(x\)- and \(y\)-intercepts, the maxima and minima, and where it is increasing and decreasing using interval notation.
Section 2.3 Characteristics of a Function’s Graph (FN3)
Subsection 2.3.1 Activities
Remark 2.3.1.
In this section, we will be looking at different kinds of graphs and will identify various characteristics. These ideas can span all kinds of functions, so you will see these come up multiple times!
Definition 2.3.2.
One of the easiest things to identify from a graph are the intercepts, which are points at which the graph crosses the axes. An \(x\)-intercept is a point at which the graph crosses the \(x\)-axis and a \(y\)-intercept is a point at which the graph crosses the \(y\)-axis. Because intercepts are points, they are typically written as an ordered pair: \((x,y)\text{.}\)
Activity 2.3.3.
(a)
What are the \(x\)-intercept(s) of \(f(x)\text{?}\)
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\(\displaystyle (0, -4)\)
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\(\displaystyle (-2, 0)\)
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\(\displaystyle (-4, 0)\)
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\(\displaystyle (0, -2)\)
(b)
What are the \(x\)-intercept(s) of \(g(x)\text{?}\)
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\(\displaystyle (0, -3)\)
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\(\displaystyle (-1, 0)\)
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\(\displaystyle (3, 0)\)
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\(\displaystyle (-3, 0)\)
(c)
What are the \(y\)-intercept(s) of \(f(x)\text{?}\)
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\(\displaystyle (0, -4)\)
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\(\displaystyle (-2, 0)\)
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\(\displaystyle (-4, 0)\)
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\(\displaystyle (0, -2)\)
(d)
What are the \(y\)-intercept(s) of \(g(x)\text{?}\)
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\(\displaystyle (0, -3)\)
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\(\displaystyle (-1, 0)\)
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\(\displaystyle (3, 0)\)
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\(\displaystyle (-3, 0)\)
(e)
Sketch a graph of a function with the following intercepts:
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\(x\)-intercepts: \((-2,0)\) and \((6,0)\)
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\(y\)-intercept: \((0,4)\)
(f)
Sketch a graph of a function with the following intercepts:
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\(x\)-intercept: \((-1,0)\)
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\(y\)-intercept: \((0,6)\) and \((0,-2)\)
Remark 2.3.4.
Notice in Activity 2.3.3, that a function can have multiple \(x\)-intercepts, but only one \(y\)-intercept. Having more than one \(y\)-intercept would create a graph that is not a function!
Definition 2.3.5.
The domain refers to the set of possible input values and the range refers to the set of possible output values. If given a graph, however, it would be impossible to list out all the values for the domain and range so we use interval notation to represent the set of values.
Aside
Activity 2.3.6.
Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
Draw on the \(x\)-axis all the values in the domain.
(b)
What interval represents the domain you drew in part (a)?
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\(\displaystyle [4, -4]\)
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\(\displaystyle [-4, 4]\)
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\(\displaystyle (-4, 4)\)
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\(\displaystyle (4, -4)\)
(c)
Draw on the \(y\)-axis all the values in the range.
(d)
What interval represents the range you drew in part (c)?
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\(\displaystyle (-5, 4)\)
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\(\displaystyle [-4, 4]\)
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\(\displaystyle [-5, 4]\)
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\(\displaystyle (4, -5)\)
Activity 2.3.8.
Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
What is the domain of this graph?
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\(\displaystyle [4, \infty)\)
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\(\displaystyle (-\infty, 0]\)
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\(\displaystyle (-\infty, 4]\)
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\(\displaystyle [0, \infty)\)
(b)
What is the range of this graph?
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\(\displaystyle [4, \infty)\)
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\(\displaystyle (-\infty, 0]\)
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\(\displaystyle (-\infty, 4]\)
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\(\displaystyle [0, \infty)\)
Remark 2.3.10.
When writing your intervals for domain and range, notice that you will need to write them from the smallest values to the highest values. For example, we wouldn’t write \([4,-\infty)\) as an interval because \(-\infty\) is smaller than \(4\text{.}\)
For domain, read the graph from left to right. For range, read the graph from bottom to top.
Activity 2.3.11.
Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
What is the domain of this graph?
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\(\displaystyle (-\infty, 3)\)
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\(\displaystyle (\infty, -4]\)
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\(\displaystyle (-4, \infty)\)
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\(\displaystyle (-\infty, 3]\)
(b)
What is the range of this graph?
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\(\displaystyle (-\infty, 3)\)
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\(\displaystyle (\infty, -4]\)
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\(\displaystyle (-4, \infty)\)
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\(\displaystyle (-\infty, 3]\)
Activity 2.3.13.
Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
What is the domain of this graph?
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\(\displaystyle (-3, 5)\)
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\(\displaystyle (-5, 7)\)
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\(\displaystyle [-5, 7]\)
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\(\displaystyle [-3,5)\)
(b)
What is the range of this graph?
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\(\displaystyle (-3, 5)\)
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\(\displaystyle (-5, 6)\)
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\(\displaystyle [-5, 6]\)
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\(\displaystyle [-3,5)\)
Remark 2.3.15.
Notice that finding the domain and range can be tricky! Be sure to pay attention to the \(x\)- and \(y\)-values of the entire graph - not just the endpoints!
Activity 2.3.16.
In this activity, we will look at where the function is increasing and decreasing. Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
Where do you think the graph is increasing?
Answer.
The function is increasing from \((-\infty, -1)\text{.}\) Instructors can emphasize this by having students think about where the function is "going up." It may be helpful to also note that the function is no longer "going up" once you get to the "top". This could help students think about how to write their answer in interval notation (with parentheses and not brackets).
(b)
Which interval best represents where the function is increasing?
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\(\displaystyle (-\infty, -1]\)
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\(\displaystyle (-\infty, -1)\)
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\(\displaystyle (-1,\infty)\)
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\(\displaystyle [-1,\infty)\)
(c)
Where do you think the graph is decreasing?
Answer.
The function is decreasing from \((-1, \infty)\text{.}\) Instructors can emphasize this by having students think about where the function is "going down." It may be helpful to also note that the function is no longer "going down" once you get to the "bottom". In this case, the "bottom" does not exist...This might be a good opportunity to discuss with students how to address this when writing the range in interval notation.
(d)
Which interval best represents where the function is decreasing?
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\(\displaystyle (-\infty, -1]\)
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\(\displaystyle (-\infty, -1)\)
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\(\displaystyle (-1,\infty)\)
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\(\displaystyle [-1,\infty)\)
(e)
Based on what you see on the graph, do you think this graph has any maxima or minima?
Definition 2.3.17.
As you noticed in Activity 2.3.16, functions can increase or decrease (or even remain constant!) for a period of time. The interval of increase is when the \(y\)-values of the function increase as the \(x\)-values increase. The interval of decrease is when the \(y\)-values of the function decrease as the \(x\)-values increase. The function is constant when the \(y\)-values remain constant as \(x\)-values increase (also known as the constant interval).
The easiest way to identify these intervals is to read the graph from left to right and look at what is happening to the \(y\)-values.
Definition 2.3.18.
The maximum, or global maximum, of a graph is the point where the \(y\)-coordinate has the largest value. The minimum, or global minimum is the point on the graph where the \(y\)-coordinate has the smallest value.
Graphs can also have local maximums and local minimums. A local maximum point is a point where the function value (i.e, \(y\)-value) is larger than all others in some neighborhood around the point. Similarly, a local minimum point is a point where the function value (i.e, \(y\)-value) is smaller than all others in some neighborhood around the point.
Remark 2.3.19.
Global extrema are sometimes referred to as absolute extrema, while local extrema are sometimes referred to as relative extrema.
Activity 2.3.20.
Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
At what value of \(x\) is there a global maximum?
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\(\displaystyle x=-4\)
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\(\displaystyle x=-3\)
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\(\displaystyle x=2\)
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\(\displaystyle x=5\)
(b)
What is the global maximum value?
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\(\displaystyle 10\)
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\(\displaystyle 8\)
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\(\displaystyle 4\)
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\(\displaystyle -5\)
(c)
At approximately what value of \(x\) is there a global minimum?
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\(\displaystyle x\approx -4\)
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\(\displaystyle x\approx -3\)
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\(\displaystyle x\approx 2\)
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\(\displaystyle x\approx 5\)
(d)
What is the global minimum value?
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\(\displaystyle 10\)
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\(\displaystyle 8\)
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\(\displaystyle 4\)
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\(\displaystyle -5\)
(e)
At approximately what value of \(x\) is there a local maximum?
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\(\displaystyle x \approx -4\)
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\(\displaystyle x \approx -3\)
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\(\displaystyle x \approx 2\)
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\(\displaystyle x \approx 5\)
(f)
What is the local maximum value?
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\(\displaystyle 10\)
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\(\displaystyle 8\)
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\(\displaystyle 4\)
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\(\displaystyle -5\)
(g)
At approximately what value of \(x\) is there a local minimum?
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\(\displaystyle x \approx -4\)
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\(\displaystyle x \approx -3\)
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\(\displaystyle x \approx 2\)
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\(\displaystyle x \approx 5\)
(h)
What is the local minimum value?
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\(\displaystyle 10\)
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\(\displaystyle 8\)
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\(\displaystyle 4\)
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\(\displaystyle -5\)
Remark 2.3.22.
Notice that in Activity 2.3.20, there are two ways we talk about max and min. We might want to know the location of where the max or min are (i.e., determining at which \(x\)-value the max or min occurs at) or we might want to know what the max or min values are (i.e., the \(y\)-value).
Also, note that in Activity 2.3.20, a local minimum is also a global minimum.
Activity 2.3.23.
Sometimes, it is not always clear what the maxima or minima are or if they exist. Consider the following graph of \(f(x)\text{:}\)
Diagram Exploration Keyboard Controls
(a)
What is the value of \(f(0)\text{?}\)
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\(\displaystyle 1\)
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\(\displaystyle 0\)
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\(f(0)\) does not exist
(b)
What is the local minimum value of \(f(x)\text{?}\)
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\(\displaystyle 1\)
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\(\displaystyle 0\)
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There is no local minimum
(c)
What is the global minimum value of \(f(x)\text{?}\)
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\(\displaystyle 1\)
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\(\displaystyle 0\)
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There is no global minimum
Activity 2.3.24.
Use the following graph to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
What is the domain?
(b)
What is the range?
(c)
What is the \(x\)-intercept(s)?
(d)
What is the \(y\)-intercept?
(e)
Where is the function increasing?
(f)
Where is the function decreasing?
(g)
Where is the constant interval?
(h)
At what \(x\)-values do the local maxima occur?
(i)
At what \(x\)-values do the local minima occur?
(j)
What are the global max and min?
Subsection 2.3.2 Exercises
Exercises available at
https://tbil.org/preview/precalculus/exercises/#/bank/FN3/
