To solve question 8, which asks to find the values of $a$ and $k$ for the equation of a parabola $y = a(x - 1)^2 + k$ given the points $(-1, 13)$ and $(2, 7)$, we substitute these points into the equation to create a system of equations:

Step 1: Substitute $(-1, 13)$ into the equation

$13 = a(-1 - 1)^2 + k$ $13 = a(2)^2 + k$ $13 = 4a + k \quad \text{(Equation 1)}$

Step 2: Substitute $(2, 7)$ into the equation

$7 = a(2 - 1)^2 + k$ $7 = a(1)^2 + k$ $7 = a + k \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

From Equation 2: $k = 7 - a$

Substitute $k = 7 - a$ into Equation 1: $13 = 4a + (7 - a)$ $13 = 4a + 7 - a$ $13 = 3a + 7$ $3a = 6$ $a = 2$

Substitute $a = 2$ back into $k = 7 - a$: $k = 7 - 2$ $k = 5$

Final Answer:

$a = 2, \, k = 5$

The equation of the parabola is: $y = 2(x - 1)^2 + 5$

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:

1. How do the values of $a$ and $k$ affect the graph of a parabola?
2. Can you derive $a$ and $k$ if only one point and the vertex are given?
3. How do you identify the vertex of this parabola from its equation?
4. What would happen to the equation if the parabola opens downward?
5. How does substituting points into a parabola's equation verify its accuracy?

Tip: When solving systems of equations, substitution is often the quickest way when one variable is easily isolated.