The linear parent function f(x)=x
 is graphed on a coordinate plane. The graph of p(x)
 represents a transformation of f(x)
. Function ​​p(x)
 has a slope that is less steep than f(x)
 and is shifted down 2
 units and right 5
 units.

Which of the following represents linear function p(x)
?
A.p(x)=−23(x−2)+5

B.p(x)=2x+5

C.p(x)=0.75(x−5)−2

D.p(x)=−5x−2


We are given that the function \( p(x) \) is a transformation of the linear parent function \( f(x) = x \), and we need to account for several transformations:

1. **Less steep slope**: The slope of \( p(x) \) must be less than 1 (the slope of \( f(x) \)).
2. **Shift down by 2 units**: This will subtract 2 from the entire function.
3. **Shift right by 5 units**: This will involve adjusting the \( x \)-term in the function.

To represent these transformations, we can use the general form of a linear function:
\[
p(x) = m(x - h) + k
\]
where:
- \( m \) is the new slope (which should be less than 1),
- \( h \) is the horizontal shift (in this case, 5 units to the right, so \( h = 5 \)),
- \( k \) is the vertical shift (in this case, 2 units down, so \( k = -2 \)).

The correct form of the function should look like this:
\[
p(x) = m(x - 5) - 2
\]
where \( m \) is a slope less than 1.

Let's analyze the options:

- **Option A**: \( p(x) = -\frac{2}{3}(x - 2) + 5 \)
    - Slope: \( -\frac{2}{3} \) (which is less than 1, but the shifts are not correct; it shifts left by 2 and up by 5).
    - This is incorrect.
    
- **Option B**: \( p(x) = 2x + 5 \)
    - Slope: 2 (which is greater than 1).
    - This is incorrect since the slope is steeper.

- **Option C**: \( p(x) = 0.75(x - 5) - 2 \)
    - Slope: 0.75 (less steep than 1).
    - Shifts: Right by 5 units (correct), and down by 2 units (correct).
    - This is the correct function.

- **Option D**: \( p(x) = -5x - 2 \)
    - Slope: -5 (much steeper, but negative).
    - This is incorrect.

Thus, the correct answer is:

**C. \( p(x) = 0.75(x - 5) - 2 \)**.

Do you want further details or have any questions?

### Related Questions:
1. What would the equation be if the function were shifted left instead of right?
2. How would the equation change if the slope were steeper instead of less steep?
3. How do vertical shifts affect the y-intercept of the function?
4. What is the effect of a negative slope on the graph's orientation?
5. How can we confirm that a function has been shifted right by 5 units?

**Tip**: Always check how both the slope and transformation terms affect the overall behavior of the function when dealing with transformations.

The linear parent function f(x)=x is graphed on a coordinate plane. The graph of p(x) represents a transformation of f(x). Function p(x) has a slope that is less steep than f(x) and is shifted down 2 units and right 5 units. Which of the following represents linear function p(x)? A.p(x)=−23(x−2)+5, B.p(x)=2x+5, C.p(x)=0.75(x−5)−2, D.p(x)=−5x−2

Learn how to solve a linear function transformation where the parent function f(x)=x is transformed to p(x) with a slope of 0.75, shifted down by 2 units and right by 5 units. This problem covers linear functions, transformations, and slope analysis. Suitable for students in Grades 8-10.

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