## 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)=−x^{2}+3

.

**Domain:** all real numbers

To find the range of q(x) :

x^{2}≥0

−x^{2}≤0

−x^{2}+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)=x^{2}−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

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

### Concept Summary

### 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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