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(x)=32(3x-1)
b f(x)=-32(x-3)
c f(x)=-2(32x-3)
d f(x)=16(2x-1)

Answer 1

To determine which function represents the linear function \( f \) given the points \( (-2, -224) \), \( (1, 64) \), \( (5, 448) \), and \( (10, 928) \), we can first analyze the points to see if there is a consistent pattern and then test each of the provided potential functions.

We first calculate the slope between two points to see if \( f \) fits a linear function model:

Using points \( (1, 64) \) and \( (5, 448) \):

\[
\text{slope} = \frac{f(5) - f(1)}{5 - 1} = \frac{448 - 64}{5 - 1} = \frac{384}{4} = 96
\]

Next, check if the values of the function at the given points fall rigidly on a linear function or confirm a different pattern.

Now, let's test each of the proposed functions to see which one matches the values of \( f(x) \):

1. **For option a: \( f(x) = 32(3x - 1) \)**

\[
f(-2) = 32(3(-2) - 1) = 32(-6 - 1) = 32(-7) = -224
\]
\[
f(1) = 32(3(1) - 1) = 32(3 - 1) = 32(2) = 64
\]
\[
f(5) = 32(3(5) - 1) = 32(15 - 1) = 32(14) = 448
\]
\[
f(10) = 32(3(10) - 1) = 32(30 - 1) = 32(29) = 928
\]

This function fits all points.

2. **For option b: \( f(x) = -32(x - 3) \)**

\[
f(-2) = -32(-2 - 3) = -32(-5) = 160 \quad (\text{not } -224)
\]

This function does not fit.

3. **For option c: \( f(x) = -2(32x - 3) \)**

\[
f(-2) = -2(32(-2) - 3) = -2(-64 - 3) = -2(-67) = 134 \quad (\text{not } -224)
\]

This function does not fit.

4. **For option d: \( f(x) = 16(2x - 1) \)**

\[
f(-2) = 16(2(-2) - 1) = 16(-4 - 1) = 16(-5) = -80 \quad (\text{not } -224)
\]

This function does not fit.

After checking all options, we see that all test results for the first function \( f(x) = 32(3x - 1) \) are consistent with the points provided.

Thus, the function that represents \( f \) is:

**a) \( f(x) = 32(3x - 1) \)**.