Question

The table represents a quadratic function f(x).


x f(x)
−10 24
−9 17
−8 12
−7 9
−6 8
−5 9
−4 12

If the equation of the function f(x) is written in standard form f(x) = ax2 + bx + c, what is the value of b?
−7
6
12
36

Answer 1

To determine the value of \( b \) in the standard form of the quadratic function \( f(x) = ax^2 + bx + c \), we can use the properties of quadratic functions and the given values in the table.

First, let's observe the values of \( f(x) \):

- \( f(-10) = 24 \)
- \( f(-9) = 17 \)
- \( f(-8) = 12 \)
- \( f(-7) = 9 \)
- \( f(-6) = 8 \)
- \( f(-5) = 9 \)
- \( f(-4) = 12 \)

Next, we can notice that \( f(x) \) has a vertex since it is quadratic, and the values drop to a minimum and then rise again. This suggests that the function is symmetrical.

To find the vertex, we can look for the average of the x-values at which the function reaches its minimum. Observing the values, the minimum seems to occur around \( x = -6 \) or \( x = -7 \).

Next, we can compute the differences between successive \( f(x) \) values to analyze the behavior of the function:

- From \( x = -10 \) to \( x = -9 \): \( 24 - 17 = 7 \)
- From \( x = -9 \) to \( x = -8 \): \( 17 - 12 = 5 \)
- From \( x = -8 \) to \( x = -7 \): \( 12 - 9 = 3 \)
- From \( x = -7 \) to \( x = -6 \): \( 9 - 8 = 1 \)
- From \( x = -6 \) to \( x = -5 \): \( 8 - 9 = -1 \)
- From \( x = -5 \) to \( x = -4 \): \( 9 - 12 = -3 \)

The differences show a symmetrical pattern around the minimum. This indicates that the minimum value of \( f(x) \) is indeed at \( x = -6 \) or \( x = -7 \).

To find the coefficients \( a \), \( b \), and \( c \), we can select points to create a system of equations.

Let's choose three points, say \( (-8, 12) \), \( (-6, 8) \), and \( (-4, 12) \).

Substituting these points into the standard form \( f(x) = ax^2 + bx + c \) gives us three equations:

1. For \( (-8, 12) \):
\[
12 = a(-8)^2 + b(-8) + c \implies 12 = 64a - 8b + c
\]

2. For \( (-6, 8) \):
\[
8 = a(-6)^2 + b(-6) + c \implies 8 = 36a - 6b + c
\]

3. For \( (-4, 12) \):
\[
12 = a(-4)^2 + b(-4) + c \implies 12 = 16a - 4b + c
\]

We can solve this system of equations step-by-step.

To eliminate \( c \), we can subtract the second equation from the first, and the third from the second:
1. \( (64a - 8b + c) - (36a - 6b + c) = 12 - 8 \)
\[
28a - 2b = 4 \implies 14a - b = 2 \quad \text{(Equation 1)}
\]

2. \( (36a - 6b + c) - (16a - 4b + c) = 8 - 12 \)
\[
20a - 2b = -4 \implies 10a - b = -2 \quad \text{(Equation 2)}
\]

Now we can solve Equations 1 and 2:
From Equation 1:
\[
b = 14a - 2
\]

Substituting into Equation 2:
\[
10a - (14a - 2) = -2
\]
\[
10a - 14a + 2 = -2
\]
\[
-4a + 2 = -2
\]
\[
-4a = -4 \implies a = 1
\]

Now substitute \( a = 1 \) back into Equation 1:
\[
b = 14(1) - 2 = 12
\]

So the value of \( b \) is \( \boxed{12} \).