We are given the curve equation $y = ax^2 + bx$ and two points: $(2, 0)$ and $(4, 8)$. Using these points, we can set up a system of equations to solve for $a$ and $b$.

Step 1: Use the point $(2, 0)$

Substitute $x = 2$ and $y = 0$ into the equation $y = ax^2 + bx$:

$0 = a(2^2) + b(2)$ $0 = 4a + 2b$ $2a + b = 0 \quad \text{(Equation 1)}$

Step 2: Use the point $(4, 8)$

Substitute $x = 4$ and $y = 8$ into the equation $y = ax^2 + bx$:

$8 = a(4^2) + b(4)$ $8 = 16a + 4b$ $4a + b = 2 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

From Equation 1: $2a + b = 0$, we can express $b$ in terms of $a$: $b = -2a$

Substitute this into Equation 2: $4a + (-2a) = 2$ $2a = 2$ $a = 1$

Now, substitute $a = 1$ into $b = -2a$: $b = -2(1) = -2$

Final Answer:

The values of $a$ and $b$ are: $a = 1, \quad b = -2$

Would you like a further explanation of any of the steps or additional details?

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Here are 5 questions that expand on this concept:

1. How would the curve change if $b = 0$?
2. Can we find the vertex of the parabola $y = ax^2 + bx$?
3. How do the signs of $a$ and $b$ affect the shape of the parabola?
4. What happens if the points $(2, 0)$ and $(4, 8)$ are switched?
5. How do you determine whether the curve opens upwards or downwards?

Tip: When solving systems of equations, always try to express one variable in terms of the other to simplify the substitution process.