Questions: Evaluate f(x)=-6x^2+4x+6 at the indicated values f(1)= f(4)= f(a+1)=

Solution Steps

To evaluate the function \( f(x) = -6x^2 + 4x + 6 \) at the given values, we need to substitute each value into the function and compute the result.

1. For \( f(1) \), substitute \( x = 1 \) into the function.
2. For \( f(4) \), substitute \( x = 4 \) into the function.
3. For \( f(a+1) \), substitute \( x = a+1 \) into the function.

Given the function \( f(x) = -6x^2 + 4x + 6 \), we need to evaluate it at \( x = 1 \).

\[ f(1) = -6(1)^2 + 4(1) + 6 \]

Calculate each term:

\[ -6(1)^2 = -6 \] \[ 4(1) = 4 \] \[ 6 = 6 \]

Add the results:

\[ f(1) = -6 + 4 + 6 = 4 \]

\[ \boxed{f(1) = 4} \]

Next, we evaluate the function at \( x = 4 \).

\[ f(4) = -6(4)^2 + 4(4) + 6 \]

Calculate each term:

\[ -6(4)^2 = -6 \cdot 16 = -96 \] \[ 4(4) = 16 \] \[ 6 = 6 \]

Add the results:

\[ f(4) = -96 + 16 + 6 = -74 \]

\[ \boxed{f(4) = -74} \]

Finally, we evaluate the function at \( x = a + 1 \).

\[ f(a+1) = -6(a+1)^2 + 4(a+1) + 6 \]

First, expand \( (a+1)^2 \):

\[ (a+1)^2 = a^2 + 2a + 1 \]

Substitute this back into the function:

\[ f(a+1) = -6(a^2 + 2a + 1) + 4(a+1) + 6 \]

Distribute the constants:

\[ f(a+1) = -6a^2 - 12a - 6 + 4a + 4 + 6 \]

Combine like terms:

\[ f(a+1) = -6a^2 - 8a + 4 \]

\[ \boxed{f(a+1) = -6a^2 - 8a + 4} \]

Final Answer