To match the equations with the graphs, let's analyze both the graphs and the equations:

Given Equations:

• A. $f(x) = (x + 4)^2 - 1$: This is a parabola with its vertex at $(-4, -1)$. Since $(x + 4)$ shifts the graph left by 4 units and $-1$ shifts it down by 1 unit.
• C. $f(x) = (x - 4)^2 - 1$: This parabola has its vertex at $(4, -1)$, shifted right by 4 units and down by 1 unit.
• E. $f(x) = 2(x - 4)^2 - 1$: This parabola also has its vertex at $(4, -1)$, but is vertically stretched because of the coefficient 2, making it narrower compared to the others.

Graphs:

• Graph B: This graph has its vertex at $(-4, -1)$ and opens upwards. This corresponds to Equation A $f(x) = (x + 4)^2 - 1$.
• Graph D: This graph has its vertex at $(4, -1)$ and opens upwards without any stretch. This corresponds to Equation C $f(x) = (x - 4)^2 - 1$.
• Graph F: This graph has its vertex at $(4, -1)$ but is narrower, indicating a vertical stretch. This corresponds to Equation E $f(x) = 2(x - 4)^2 - 1$.

Final Matches:

• A. $f(x) = (x + 4)^2 - 1$ matches with Graph B.
• C. $f(x) = (x - 4)^2 - 1$ matches with Graph D.
• E. $f(x) = 2(x - 4)^2 - 1$ matches with Graph F.

Would you like more details on how transformations of parabolas work, or do you have any further questions?

Here are five related questions for further practice:

1. How does changing the sign of the coefficient of $x^2$ affect the parabola?
2. What happens to the parabola when the constant term is increased or decreased?
3. How does the value of the coefficient in front of $(x - h)^2$ affect the width of the parabola?
4. How would the graph change if the equation was $f(x) = (x - 4)^2 + 3$?
5. Can you determine the axis of symmetry for each of these equations?

Tip: The vertex form of a quadratic equation, $f(x) = a(x - h)^2 + k$, makes it easy to identify the vertex as $(h, k)$ and the factor $a$ controls the width and direction of the parabola.