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# Analyzing Functions Graphically

## Key Concepts

• Identify the common features of a function when given an equation or graph.
• Analyze domain and range of the function.
• Analyze maximum and minimum values of the function.
• Understand axes of symmetry of the function.
• Analyze end behaviors of the graphs.

### Graph of the functions and its features

Q 1: Plot the points of each function on a graph. Explain the features of the function that represents the graph:

Solution:

Graph A:

Common features of the graph A:

Domain: [-2, 2]

Range: [1, 5]

There is no x-intercept.

There is no y-intercept.

Graph B:

Common features of the graph B:

Domain: [-2, 2]

Range: [-1, 11]

x-intercept: 1.7

y-intercept: 5

Graph C:

Common features of the graph C:

Domain: [-2, 2]

Range: [3, 21]

There is no x-intercept.

y-intercept: 5

### Analyze Domain and Range

The domain of a function is the set of all values for which the function is defined.

The range of the function is the set of all values that the function takes.

Example 1:

The graph of the function: p(x)=|x|−1.

Domain: all real numbers

To find the range of p(x):

|x| ≥ 0

|x|−1≥−1

p(x)≥−1

Range:

y≥−1

Example 2:

The graph of the function: q(x)=−x2+3

.

Domain: all real numbers

To find the range of q(x) :

x2≥0

−x2≤0

−x2+3≤3

q(x)≤3

Range: y≤3.

### Analyze Maximum and Minimum Values

Example 3:

The graph of the function: f(x)=−2x+5.

Solution:

The graph of the linear function f(x)=−2x+5 decreases at a constant rate.   So, there is no maximum or minimum value.

Example 4:

The graph of the function:

g(x)=−2x+3.

Solution:

The graph of the function g(x)=−2x+3 is a translation of an exponential function.

It is bounded above the asymptote y=3 which means that g(x)<3.

It has no maximum because it is approaching 3 but never reaches 3.

The function g also has no minimum. As x increases, g(x) decreases.

Example 5:

The graph of the function: h(x)=|x−2|−1.

Solution:

The graph of the function h(x)=|x−2|−1 is a translation of an absolute value function.

It opens upward so the function has a minimum value of – 1 at the vertex (2,−1).

### Understand Axes of Symmetry

Example 6:

The graph of the function: p(x)=5−|x+1|.

Solution:

Translations of the absolute value function always have an axis of symmetry passing through the vertex.

Here the function

p(x)=5−|x+1| has an axis of symmetry x=−1 passing through the vertex (−1, 5).

Example 7:

The graph of the function: q(x)=(x+3).

Solution:

Quadratic functions always have a vertical axis of symmetry.

Here the quadratic function q(x)=(x+3)2 has an axis of symmetry x=−3.

Example 8:

The graph of the function:

r(x)=√x+2

Solution:

The function r(x)=√x+2 does not have an axis of symmetry.

There is no way to fold the graph so that one side aligns with the other.

### Analyze End Behaviors of Graphs

Example 9:

The graph of the function:

h(x)=x2−2x+1.

Solution:

As x→∞ , the values of h(x) increases without bound.

So, h(x)→∞.

As x→−∞ , the values of h(x) also increases without bound.

So, h(x)→∞ .

Example 10:

The graph of the function:

g(x)=∛x−2.

Solution:

As x→∞ , the values of g(x) grow less and less steeply, but they do not approach to any asymptote.

So, g(x)→∞.

As x→−∞ , the values of g(x) also decreases.

So, g(x)→−∞.

## Exercise

1. Sketch the graph of the function fx=√x-4 and identify its domain and range.
2. Use the graph of the function fx=5|x|-8 to identify its maximum and minimum value if they exist.
3. Describe the end behavior of the function fx=-7x.

### What we have learned

• Identify the common features of a function when given an equation or graph.
• Analyze domain and range of the function.
• Analyze maximum and minimum values of the function.
• Understand axes of symmetry of the function.
• Analyze end behaviors of the graphs.

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