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17Mark For Review

The rational function g is defined by an equation in the form g(x)=ax+b, where a and b are constants. The partial graph of y=g(x) is shown. If h(x)=g(x+2), which equation could define function h?

h(x)=12x+2
h(x)=12x+4
h(x)=12x+6
h(x)=12xx+6

To solve the problem, let's first analyze the given graph of \( y = g(x) \), where \( g(x) = ax + b \).

### Step 1: Identify the slope and y-intercept of \( g(x) \)

Since the function \( g(x) \) is linear (in the form \( g(x) = ax + b \)), the graph will be a straight line. 

Given the grid lines in the graph:
- The slope \( a \) can be determined by the rise over run (change in \( y \) over change in \( x \)).
- The \( y \)-intercept \( b \) is the point where the line crosses the \( y \)-axis.

### Step 2: Translate the function

The function \( h(x) = g(x + 2) \) represents a horizontal shift of the function \( g(x) \) by 2 units to the left. The general form of \( g(x) \) is \( ax + b \). Applying the shift:

\[
h(x) = g(x + 2) = a(x + 2) + b = ax + 2a + b
\]

So, the new equation for \( h(x) \) is \( h(x) = ax + (2a + b) \).

### Step 3: Match with the options

From the options given:
1. \( h(x) = \frac{1}{2}x + 2 \)
2. \( h(x) = \frac{1}{2}x + 4 \)
3. \( h(x) = \frac{1}{2}x + 6 \)
4. \( h(x) = \frac{1}{2} \cdot x + 6 \) (this is the same as option 3)

We need to identify which equation matches the form \( h(x) = \frac{1}{2}x + (2a + b) \).

### Step 4: Identify the correct equation

The graph indicates the slope \( a = \frac{1}{2} \) (as the line rises 1 unit for every 2 units run). To determine \( b \), note the original y-intercept on the graph. If the intercept was 2, and \( 2a + b \) should match one of the choices.

Since \( h(x) = ax + (2a + b) \), plug in \( a = \frac{1}{2} \):

\[
2 \times \frac{1}{2} + b = 1 + b
\]

If \( 1 + b = 6 \), then \( b = 5 \). But if the intercept was 4:

\[
1 + b = 4 \quad \text{or} \quad b = 3
\]

Thus, by matching the intercept shift, the equation would likely be \( h(x) = \frac{1}{2}x + 4 \).

So, the correct equation for \( h(x) \) is:

\[
\boxed{h(x) = \frac{1}{2}x + 4}
\]

Would you like any further details or explanations?

### Follow-up Questions:
1. How can you identify the slope and y-intercept directly from the graph?
2. What does the transformation \( g(x + 2) \) do to the graph of \( g(x) \)?
3. How would the equation change if the transformation were \( g(x - 2) \)?
4. How does the form of a linear equation help in identifying its graph?
5. What other types of transformations can be applied to linear functions?

### Tip:
When dealing with transformations, always remember how the horizontal and vertical shifts affect the graph. Horizontal shifts are counterintuitive, moving the graph in the opposite direction to the sign in the equation.

Explore how to transform linear functions and analyze function shifts using the example of g(x) and h(x). Learn step-by-step how to determine the equation of h(x) when given g(x).

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