Questions: For the function f(x) = 4x^2 + 6x - 1, evaluate f(a + h).

Transcript text: For the function $f(x)=4 x^{2}+6 x-1$, evaluate $f(a+h)$.

Solution

Solution Steps

To evaluate $ f(a+h) $ for the function $ f(x) = 4x^2 + 6x - 1 $, we need to substitute $ a+h $ in place of $ x $ in the function and simplify the expression.

Step 1: Define the Function

The given function is $ f(x) = 4x^2 + 6x - 1 $.

Step 2: Substitute $ a+h $ into the Function

To find $ f(a+h) $, we substitute $ a+h $ for $ x $ in the function: \[ f(a+h) = 4(a+h)^2 + 6(a+h) - 1 \]

Step 3: Expand the Expression

Next, we expand the expression $ 4(a+h)^2 + 6(a+h) - 1 $: \[ 4(a+h)^2 = 4(a^2 + 2ah + h^2) = 4a^2 + 8ah + 4h^2 \] \[ 6(a+h) = 6a + 6h \] Combining these, we get: \[ f(a+h) = 4a^2 + 8ah + 4h^2 + 6a + 6h - 1 \]

Final Answer

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

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