The given image presents a function diagram with the rule $f(x) = ax^2 + b$. The diagram shows the mapping of values from one set to another. Specifically, it shows the following mappings:

• $f(-1) = 5$
• $f(-2) = 5$
• $f(2) = 1$

We are asked to find the value of $6a + 2b$, where $a$ and $b$ are constants from the function $f(x) = ax^2 + b$.

Step 1: Use the given mappings to set up equations

For each input-output pair in the function, substitute the values into $f(x) = ax^2 + b$ to form a system of equations.

1. $f(-1) = 5$ gives us: $a(-1)^2 + b = 5 \quad \Rightarrow \quad a + b = 5$
2. $f(-2) = 5$ gives us: $a(-2)^2 + b = 5 \quad \Rightarrow \quad 4a + b = 5$
3. $f(2) = 1$ gives us: $a(2)^2 + b = 1 \quad \Rightarrow \quad 4a + b = 1$

Step 2: Solve the system of equations

From the first two equations:

• Equation 1: $a + b = 5$
• Equation 2: $4a + b = 5$

Subtract Equation 1 from Equation 2: $(4a + b) - (a + b) = 5 - 5 \quad \Rightarrow \quad 3a = 0 \quad \Rightarrow \quad a = 0$

Substitute $a = 0$ into $a + b = 5$: $0 + b = 5 \quad \Rightarrow \quad b = 5$

Step 3: Calculate $6a + 2b$

Now that we know $a = 0$ and $b = 5$, we can substitute these into the expression $6a + 2b$: $6(0) + 2(5) = 0 + 10 = 10$

It seems none of the provided answer choices matches the calculated value of 10. However, based on the function analysis, the correct calculation yields 10.

Let me know if you'd like further clarification or if there are any specific steps you'd like to explore in more detail!

Here are 5 related questions to deepen understanding:

1. How would the solution change if the value of $a$ was not zero?
2. What does the graph of the function $f(x) = ax^2 + b$ look like based on the values of $a$ and $b$?
3. How do quadratic functions behave when $a = 0$?
4. What would the system of equations look like if we were given additional points?
5. How can we solve systems of linear equations using substitution or elimination?

Tip: Always double-check if there are multiple outputs for a single input, as it could affect the consistency of the function.