Question

The table represents some points on the graph of linear function `f`.
x = -2, 1, 5, 10
f(x) = -224, 64, 448, 928
Which function represents `f`?
a. f\left(x\right)=32\left(3x-1\right)
b. f\left(x\right)=-32\left(x-3\right)
c. f\left(x\right)=-2\left(32x-3\right)
d. f\left(x\right)=16\left(2x-1\right)

Answer 1

To find the correct function that represents \( f \), we need to determine the linear relationship between \( x \) and \( f(x) \) using the given points:

- \( (-2, -224) \)
- \( (1, 64) \)
- \( (5, 448) \)
- \( (10, 928) \)

First, we can calculate the slope \( m \) of the linear function \( f(x) = mx + b \) using any two points. Let's use the points \( (1, 64) \) and \( (5, 448) \):

The slope \( m \) is given by:
\[
m = \frac{f(5) - f(1)}{5 - 1} = \frac{448 - 64}{5 - 1} = \frac{384}{4} = 96
\]

Now we have the slope. Next, we can use the point-slope form of the line to find the y-intercept \( b \). Let's use the point \( (1, 64) \):

Using the slope-intercept form:
\[
f(x) = mx + b
\]
we get:
\[
64 = 96(1) + b
\]
\[
64 = 96 + b
\]
\[
b = 64 - 96 = -32
\]

Thus, the function can be represented as:
\[
f(x) = 96x - 32
\]

Now, let's check the function with the provided options. To do that, we can simplify our equation \( f(x) = 96x - 32 \) to see if it matches any of the options given.

Now, let's evaluate the provided options:

1. **Option a: \( f(x) = 32(3x - 1) \)**
- Simplifying: \( f(x) = 96x - 32 \) (matches!)

2. **Option b: \( f(x) = -32(x - 3) \)**
- Simplifying: \( f(x) = -32x + 96 \) (does not match)

3. **Option c: \( f(x) = -2(32x - 3) \)**
- Simplifying: \( f(x) = -64x + 6 \) (does not match)

4. **Option d: \( f(x) = 16(2x - 1) \)**
- Simplifying: \( f(x) = 32x - 16 \) (does not match)

The only option that matches our derived function \( f(x) = 96x - 32 \) is option **a**.

Thus, the correct answer is:
\[ \boxed{a} \]