1. The problem asks to match each quadratic equation with its correct graph and then sketch graphs of given quadratic relations by applying transformations to the base graph $y = x^2$. 2. Quadratic equations in vertex form are $y = (x - h)^2 + k$, where $(h,k)$ is the vertex. 3. For each equation, identify the vertex: - a) $y = (x - 2)^2 + 3$ has vertex $(2,3)$ - b) $y = (x + 2)^2 - 3$ has vertex $(-2,-3)$ - c) $y = (x + 3)^2 - 2$ has vertex $(-3,-2)$ - d) $y = (x - 3)^2 + 2$ has vertex $(3,2)$ 4. Match vertices to graph positions: - Top-left: vertices with negative $x$ and positive $y$ or positive $y$ near left - Bottom-left: vertices with negative $x$ and negative $y$ - Top-right: vertices with positive $x$ and positive $y$ - Bottom-right: vertices with positive $x$ and negative $y$ 5. Assign graphs: - a) $(2,3)$ top-right (v) - b) $(-2,-3)$ bottom-left (ii) - c) $(-3,-2)$ bottom-left (iv) - d) $(3,2)$ top-right (v) but since v is taken, d) matches (vi) bottom-right is positive $x$ but positive $y$, so better (vi) bottom-right 6. Sketch graphs by starting with $y = x^2$ and applying transformations: - a) $y = x^2 - 4$: shift down 4 units - b) $y = (x - 3)^2$: shift right 3 units - c) $y = x^2 + 2$: shift up 2 units - d) $y = (x + 5)^2$: shift left 5 units - e) $y = (x + 1)^2 - 2$: shift left 1 unit and down 2 units - f) $y = (x - 5)^2 + 3$: shift right 5 units and up 3 units Final answers: **Matching:** - a) (v) - b) (ii) - c) (iv) - d) (vi) **Sketch transformations:** - a) down 4 - b) right 3 - c) up 2 - d) left 5 - e) left 1, down 2 - f) right 5, up 3