If g(x) is the f(x)=x
 after a vertical compression by 1/2
 
, a shift left by 1, and a shift up by 3,                                  a)write an equation for g(x)
  

b) The slope of this line is: 
  

c) The vertical intercept of this line is:

Let's solve this step by step.

### a) Write an equation for \( g(x) \).

Given the function \( f(x) = x \), let's apply the transformations:

1. **Vertical compression by \( \frac{1}{2} \)**: This transformation scales the function vertically by \( \frac{1}{2} \). So, the new function becomes:
   \[
   f_1(x) = \frac{1}{2}x
   \]

2. **Shift left by 1**: Shifting the function left by 1 unit means replacing \( x \) with \( x + 1 \). The function becomes:
   \[
   f_2(x) = \frac{1}{2}(x + 1)
   \]

3. **Shift up by 3**: Shifting the function up by 3 units means adding 3 to the function. The final function is:
   \[
   g(x) = \frac{1}{2}(x + 1) + 3
   \]

Simplifying this, we get:
\[
g(x) = \frac{1}{2}x + \frac{1}{2} + 3
\]
\[
g(x) = \frac{1}{2}x + \frac{7}{2}
\]

### b) The slope of this line is:

The slope of the line is the coefficient of \( x \) in the equation for \( g(x) \). From the equation \( g(x) = \frac{1}{2}x + \frac{7}{2} \), the slope \( m \) is:
\[
\text{slope} = \frac{1}{2}
\]

### c) The vertical intercept of this line is:

The vertical intercept (also known as the y-intercept) is the constant term in the equation for \( g(x) \). In this case, the y-intercept is:
\[
\text{vertical intercept} = \frac{7}{2} = 3.5
\]

### Summary:
- **a)** The equation for \( g(x) \) is \( g(x) = \frac{1}{2}x + \frac{7}{2} \).
- **b)** The slope of the line is \( \frac{1}{2} \).
- **c)** The vertical intercept is \( \frac{7}{2} \) or 3.5.

Would you like further details or have any other questions?

Here are some related questions:
1. How would the equation change if the function was reflected over the x-axis?
2. What would the equation be if the function was shifted right by 2 units instead of left by 1 unit?
3. How would the slope change if the function had a vertical stretch by a factor of 3?
4. What is the equation for \( g(x) \) if the function was shifted down by 4 units instead of up by 3 units?
5. How would the y-intercept change if the slope were doubled?

**Tip:** Always apply transformations in the correct order: stretches/compressions first, then shifts.

If g(x) is the f(x) = x after a vertical compression by 1/2, a shift left by 1, and a shift up by 3, a) write an equation for g(x). b) The slope of this line is: c) The vertical intercept of this line is:

Learn how to transform the linear function f(x) = x by applying a vertical compression, a left shift, and an upward shift. This detailed step-by-step solution helps students understand transformations, determine the slope, and find the vertical intercept. Suitable for Grades 8-10.

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