To find the function \( G(x) \) derived from the given function \( F(x) \), we first need to correctly interpret and define \( F(x) \) based on the equation you've provided.
Given the equation:
\[ \frac{1}{2} x = -12x^2 + 21x - 3 \]
To express it in a standard form, we will eliminate \(\frac{1}{2} x\) from one side:
\[ -12x^2 + 21x - 3 - \frac{1}{2}x = 0 \]
To combine the \( x \) terms, we can write \( 21x - \frac{1}{2}x \) as:
\[ 21x - 0.5x = 20.5x \]
So, we have:
\[ -12x^2 + 20.5x - 3 = 0 \]
This means \( F(x) = -12x^2 + 20.5x - 3 \).
Creating \( G(x) \):
If we want to create a function \( G(x) \) by horizontally stretching \( F(x) \), we replace \( x \) in \( F(x) \) with \( kx \), where \( k > 1 \). The horizontal stretching is affected by the factor \( k \).
If the k factor is 1/2, you will actually be horizontally compressing the function. To apply a horizontal stretch of factor \( k \), our new function becomes:
\[ G(x) = F(kx) \]
In this case, if the stretch factor was incorrectly stated and we actually meant a horizontal compression by a factor of 1/2, we would substitute \( x \) with \( 2x \):
\[ G(x) = F(2x) \]
Let’s proceed with substituting \( 2x \) into \( F(x) \):
\[ G(x) = -12(2x)^2 + 20.5(2x) - 3 \] \[ = -12(4x^2) + 41x - 3 \] \[ = -48x^2 + 41x - 3 \]
Creating Table 2:
Now, we can create Table 2 by calculating \( G(x) \) for values of \( x \). Using the given values \( x = 0, 1, 2 \):
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For \( x = 0 \): \[ G(0) = -48(0)^2 + 41(0) - 3 = -3 \]
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For \( x = 1 \): \[ G(1) = -48(1)^2 + 41(1) - 3 = -48 + 41 - 3 = -10 \]
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For \( x = 2 \): \[ G(2) = -48(2)^2 + 41(2) - 3 = -48(4) + 82 - 3 = -192 + 82 - 3 = -113 \]
So your table two would look like this:
| \( x \) | \( G(x) \) | |---------|------------| | 0 | -3 | | 1 | -10 | | 2 | -113 |
This summarizes the calculations for \( G(x) \) based on the original function \( F(x) \). Let me know if there are further details you'd like me to explore!