Question

Find \(a\) so that the point (-1,2) is on the graph of \(f(x)=a x^{2}+4\)

Short answer

The value of `a` is -2.

Step-by-step solution

Understand the problem

You need to find the value of the constant `a` such that when \(x = -1\) and \(y = 2\), the point satisfies the equation \(f(x) = ax^2 + 4\).

Substitute the point into the function

Substitute \(x = -1\) and \(y = 2\) into the function \(f(x) = ax^2 + 4\), which results in \(2 = a(-1)^2 + 4\).

Simplify the equation

Simplify the equation: \[ 2 = a \times 1 + 4 \] which simplifies to \[ 2 = a + 4 \].

Solve for `a`

Isolate `a` by subtracting 4 from both sides: \[ 2 - 4 = a \] which simplifies to \[ a = -2 \].

Verify the solution

Substitute \(a = -2\) back into the function to verify: \[ f(x) = -2(-1)^2 + 4 = -2 + 4 = 2 \]. The point (-1,2) is indeed on the graph.

Key concepts

Substituting Values

Substituting values means replacing the variables in an equation with specific numbers to solve for unknowns. In our exercise, we had the function
\(f(x) = ax^2 + 4\) and the point (-1, 2) which needed to lie on this graph.
To check this, we substituted
\(x = -1\) and \(y = 2\) into the function. This resulted in the equation
\(2 = a(-1)^2 + 4\).
The substitution step is crucial because it allows you to transform the function into an equation that involves only one variable, `a`, making it easier to solve.
By doing so, we can systematically approach finding the value of `a`.

Isolating Variables

Isolating variables is the process of rearranging an equation to get the unknown variable by itself on one side of the equation. In our case, after substituting the values, we obtained the equation
\(2 = a \times 1 + 4\).
Let's simplify the equation to make it more straightforward. Simplification gave us
\(2 = a + 4\).
To isolate `a`, we need to remove the 4 from the right-hand side. We do this by subtracting 4 from both sides of the equation:
\(2 - 4 = a\).
This resulted in
\(a = -2\).
Isolating the variable `a` helped us determine its value accurately by ensuring it was the only term on one side of the equation.
This step is vital as it directs us towards a clear and precise solution.

Verifying Solutions

Verifying solutions is the final step where you check if the obtained solution satisfies the original equation. In this problem, we found that
\(a = -2\).
However, it's essential to verify this solution to ensure its correctness.
To do this, we substituted
\(a = -2\) back into our original function:
\[f(x) = -2(-1)^2 + 4\]
The calculation results are:
\[f(x) = -2 \times 1 + 4 = -2 + 4 = 2\]
Since the result is 2, we confirm that the point (-1, 2) satisfies the equation with
\(a = -2\).
Verifying solutions is essential because it ensures that the value found for the variable correctly fits the initial conditions.
This step guarantees the accuracy and validity of the solution found.