## Key Concepts

- Move the graph of an exponential function vertically.
- Explain about the horizontal translation of an exponential graph.
- Compare two different transformations of f(x)=2
^{x }

### Transformation of Exponential Functions

#### 1. Quadratic function

A function f defined by f(x)=ax^{2}+bx+c, where a, b, and c are real numbers and a≠0 is called a *quadratic function*.

- The graph of a quadratic function is a curve called a
**parabola**.

- The quadratic parent function is f(x)=x
^{2} - For 0<|a|<1, the shape of the parabola is wider than the parent function.
- For |a|>1, the shape of the parabola is narrower than the parent function.

- f(x)=ax
^{2}is the reflection of f(x)=−ax^{2}over the x-axis.

#### 2. Graph of g(x) = x^{2}+h

- The value of kk in g(x)=x
^{2}+k=translates the graph of parent function f, vertically k units. - The value of k does not affect the axis of symmetry.

#### 3. Graph of (x−h)^{2}

- The value of h in g(x)=(x−h)
^{2}translates the graph of parent function f, horizontally h units. - The vertex of the graph g is (0, h).
- The value of h translates the axis of symmetry.

#### 4. Exponential function

- The product of an initial amount and a constant ratio raised to a power is an
**exponential function**. - Exponential functions are modeled using f(x)=a.b
^{x}where a is a non-zero constant, b>0 and b≠1.

1. **Vertical translations of graphs of exponential functions**

The graph of f(x)=2^{x}+k is a vertical translation of the graph of f(x)=2^{x }

If k is positive, the graph is moved up.

If k is negative, the graph is moved down.

**Example:** Compare the graph of f(x)=a^{x}−k with the parent function.

The graph moves downwards.

**Example: **Compare the graph of f(x)=a^{x}+k

The graph moves upwards.

2. **Horizontal translations of graphs of exponential functions**

The graph of f(x)=2^{x}−h is a horizontal translation of the graph of f(x)=2^{x}

- If hh is positive, the graph is translated to the right.
- If hh is negative, the graph is translated to the left.

**Example:** Compare the graph of f(x)=a^{x}−h with the parent function when h>0.

The graph moves to the right by h units.

**Example:** Compare the graph of f(x)=a^{x}−h with the parent function when h<0.

The graph moves to the left by h units.

3. **Compare two different transformations of **f(x)=2^{x}

- We can compare two different transformations of f(x)=2
^{x}

- Compare the asymptote, y-intercept of each transformation function (from the function or the graph of the function) with respect to the f(x)
- Identify the difference between the asymptotes of the two transformation functions.

## Exercise

- How does the graph of g(x)=2^x+1 compare to the graph of f(x)=2^x?
- Compare the graph of f(x)=2^(x+2) with the graph of f(x)=2^x.
- How does the graph of m(x)=3^x-4 compare to the graph of p(x)=3^x+7?
- Compare the function represented by the graph of g(x)=2^x-3 to the function represented by the table.

- Find the value of
*k*from the graph.

### Concept Map

### What we have learned

- The graph of f(x)=2^x+k is a vertical translation of the graph of f(x)=2^x.
- The graph of f(x)=2^(x-h) is a horizontal translation of the graph of f(x)=2^x.

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