Let's go through each question in the image step by step.

1. What is $f(-1)$ if $f(x) = -x^3 - x^2 + 2x - 7$?

To find $f(-1)$, substitute $x = -1$ into the function:

$f(-1) = -(-1)^3 - (-1)^2 + 2(-1) - 7$ $f(-1) = 1 - 1 - 2 - 7 = -9$

Answer: B. $f(-1) = -9$

2. Evaluate the given function if $x = a - 1$, $f(x) = 2x^2 - 9$?

Substitute $x = a - 1$ into the function:

$f(a-1) = 2(a-1)^2 - 9$ $f(a-1) = 2(a^2 - 2a + 1) - 9$ $f(a-1) = 2a^2 - 4a + 2 - 9 = 2a^2 - 4a - 7$

Answer: A. $f(a-1) = 2a^2 - 4a - 7$

3. What is $k(-b)$ if $k(x) = 2x^2 - x - 9$?

Substitute $x = -b$ into the function:

$k(-b) = 2(-b)^2 - (-b) - 9$ $k(-b) = 2b^2 + b - 9$

Answer: D. $2b^2 + b - 9$

4. A projectile is shot upward from 5m above the ground with an initial velocity of 35 m/s. The height of the projectile, $s$, in meters after $t$ seconds is given by $s(t) = 5 + 35t - 9.8t^2$. Calculate the height after 2.5 seconds.

Substitute $t = 2.5$ into the equation:

$s(2.5) = 5 + 35(2.5) - 9.8(2.5)^2$ $s(2.5) = 5 + 87.5 - 9.8(6.25)$ $s(2.5) = 5 + 87.5 - 61.25 = 31.25 \, \text{meters}$

Answer: B. 31.25 meters

5. What is $f(1)$ if $f(x) = -x^3 - x^2 + 2x - 7$?

Substitute $x = 1$ into the function:

$f(1) = -(1)^3 - (1)^2 + 2(1) - 7$ $f(1) = -1 - 1 + 2 - 7 = -7$

Answer: B. $f(1) = -7$

Would you like further explanations or have any questions?

Here are five related questions to deepen understanding:

1. How do you evaluate polynomial functions for any given value of $x$?
2. What happens to the value of a function if $x$ is negative or positive?
3. How does the quadratic term $x^2$ affect the shape of a function's graph?
4. How can you find the maximum height of a projectile in a quadratic motion problem?
5. How would you solve the same type of function evaluation problems using a graphing approach?

Tip: Always substitute values carefully when evaluating functions, and pay special attention to signs when squaring or dealing with negative terms.