Question
Find \( a \) so that the point \( (-1,5) \) is on the graph of \( f(x)=a x^{2}+9 \). (Give an exact answer. Use symbolic notation and fractions where needed.) \( a= \) a
Ask by Marsh Joseph. in the United States
Jan 29,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( a = -4 \)
Solution
To find the value of \( a \) such that the point \( (-1, 5) \) lies on the graph of the function \( f(x) = a x^2 + 9 \), follow these steps:
1. **Substitute the Point into the Function:**
The point \( (-1, 5) \) means that when \( x = -1 \), \( f(x) = 5 \). So, substitute \( x = -1 \) and \( f(x) = 5 \) into the equation:
\[
5 = a(-1)^2 + 9
\]
2. **Simplify the Equation:**
\[
5 = a(1) + 9 \quad \Rightarrow \quad 5 = a + 9
\]
3. **Solve for \( a \):**
\[
a = 5 - 9 = -4
\]
**Answer:**
\( a = -4 \)
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Simplify this solution Extra Insights
To find \( a \) such that the point \( (-1,5) \) is on the graph of \( f(x) = a x^{2} + 9 \), we substitute \( x = -1 \) and \( f(x) = 5 \) into the equation: \[ 5 = a(-1)^{2} + 9 \] This simplifies to: \[ 5 = a(1) + 9 \] Next, we solve for \( a \): \[ 5 = a + 9 \] \[ a = 5 - 9 \] \[ a = -4 \] Thus, the value of \( a \) is \( -4 \).
