Ask a New Question
Menu

Use the tables to answer the question.

The function g(x)
is created by horizontally stretching the parent function, f(x)=−12x2+21x−3
, with a k factor of 12
. Complete the table to show how the function has changed. (1 point)
Table 1

x
f(x)
0
−3
1
6
2
−9

Table 2

x
g(x)
0
1
2

1 answer

To find the values of \( g(x) \) based on the transformation of the function \( f(x) = -12x^2 + 21x - 3 \) through a horizontal stretch by a factor of \( k = 12 \), we first need to understand how the transformation affects the input values.

A horizontal stretch by a factor of \( k \) means that if we have a function \( f(x) \), the new function \( g(x) \) can be represented as:

\[ g(x) = f\left(\frac{x}{k}\right) = f\left(\frac{x}{12}\right) \]

We will use this to calculate \( g(0) \), \( g(1) \), and \( g(2) \) based on the values of \( f(x) \) from Table 1:

Calculating the values for \( g(x) \)

  1. For \( g(0) \):

    \[ g(0) = f\left(\frac{0}{12}\right) = f(0) \]

    From Table 1: \[ f(0) = -3 \] So, \[ g(0) = -3 \]

  2. For \( g(1) \):

    \[ g(1) = f\left(\frac{1}{12}\right) \]

    To find \( f\left(\frac{1}{12}\right) \), we substitute \( x = \frac{1}{12} \) into the function \( f(x) \):

    \[ f\left(\frac{1}{12}\right) = -12\left(\frac{1}{12}\right)^2 + 21\left(\frac{1}{12}\right) - 3 \]

    Calculating each term: \[ -12\left(\frac{1}{12}\right)^2 = -12\left(\frac{1}{144}\right) = -\frac{12}{144} = -\frac{1}{12} \]

    \[ 21\left(\frac{1}{12}\right) = \frac{21}{12} = \frac{7}{4} \]

    So, \[ f\left(\frac{1}{12}\right) = -\frac{1}{12} + \frac{7}{4} - 3 \]

    Convert \(-3\) into twelfths: \[ -3 = -\frac{36}{12} \]

    Now add these together: \[ -\frac{1}{12} + \frac{21}{12} - \frac{36}{12} = \frac{-1 + 21 - 36}{12} = \frac{-16}{12} = -\frac{4}{3} \]

    Thus, \[ g(1) = -\frac{4}{3} \]

  3. For \( g(2) \):

    \[ g(2) = f\left(\frac{2}{12}\right) = f\left(\frac{1}{6}\right) \]

    Substitute \( x = \frac{1}{6} \) into function \( f(x) \):

    \[ f\left(\frac{1}{6}\right) = -12\left(\frac{1}{6}\right)^2 + 21\left(\frac{1}{6}\right) - 3 \]

    Calculating each term: \[ -12\left(\frac{1}{6}\right)^2 = -12\left(\frac{1}{36}\right) = -\frac{12}{36} = -\frac{1}{3} \]

    \[ 21\left(\frac{1}{6}\right) = \frac{21}{6} = \frac{7}{2} \]

    So, \[ f\left(\frac{1}{6}\right) = -\frac{1}{3} + \frac{21}{6} - 3 \]

    Convert \(-3\) into sixths: \[ -3 = -\frac{18}{6} \]

    Now combine: \[ -\frac{1}{3} + \frac{21}{6} - \frac{18}{6} = -\frac{2}{6} + \frac{21}{6} - \frac{18}{6} = \frac{1}{6} \]

    Thus, \[ g(2) = \frac{1}{6} \]

Finalizing Table 2

Now we can complete Table 2:

| x | g(x) | |---|-----------| | 0 | -3 | | 1 | -\(\frac{4}{3}\) | | 2 | \(\frac{1}{6}\) |

So the completed values for \( g(x) \) are:

  • \( g(0) = -3 \)
  • \( g(1) = -\frac{4}{3} \)
  • \( g(2) = \frac{1}{6} \)