If function h is defined by h(x)=`ax^(2)-7 and h(-3)=29`, what is `h((1)/(2))`?

Options

A. `-6`
B. `-5`
C. `4`
D. 8

Answer

To solve the problem step by step, we will follow these instructions:

1. **Understand the function**: We are given the function \( h(x) = ax^2 - 7 \) and the condition \( h(-3) = 29 \).

2. **Substitute the value into the function**: To find the value of \( a \), we substitute \( x = -3 \) into the function:
\[ h(-3) = a(-3)^2 - 7 = 29 \]
This simplifies to:
\[ h(-3) = 9a - 7 = 29 \]

3. **Solve for \( a \)**: Now, we will isolate \( a \):
\[ 9a - 7 = 29 \]
Adding 7 to both sides gives:
\[ 9a = 29 + 7 \]
\[ 9a = 36 \]
Dividing both sides by 9:
\[ a = \frac{36}{9} = 4 \]

4. **Substitute \( a \) back into the function**: Now that we have \( a = 4 \), we can rewrite the function:
\[ h(x) = 4x^2 - 7 \]

5. **Find \( h\left(\frac{1}{2}\right) \)**: We will now find the value of the function at \( x = \frac{1}{2} \):
\[ h\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^2 - 7 \]
This simplifies to:
\[ h\left(\frac{1}{2}\right) = 4 \cdot \frac{1}{4} - 7 \]
\[ h\left(\frac{1}{2}\right) = 1 - 7 = -6 \]

6. **Final answer**: Therefore, the value of \( h\left(\frac{1}{2}\right) \) is \( -6 \).