1. The problem asks to verify if the first two parabolas are correctly graphed and then to explain how to graph the third and fourth parabolas. 2. For the first parabola: $y = -(x + 3)^2 - 1$ - The vertex form of a parabola is $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. - Here, $a = -1$, $h = -3$, and $k = -1$. - The vertex is at $(-3, -1)$, and since $a$ is negative, the parabola opens downward. - The description says the vertex is at $(-3, -1)$ and the graph is in the top-left grid, which matches the equation. 3. For the second parabola: $y = 2x^2 - 5$ - This is in standard form but can be rewritten in vertex form as $y = 2(x - 0)^2 - 5$. - The vertex is at $(0, -5)$, and since $a = 2 > 0$, the parabola opens upward. - The description says the vertex is at $(0, -5)$ and the graph is in the top-right grid, which matches the equation. 4. Both first and second parabolas are correctly described and graphed. 5. Now, let's walk through how to graph the third parabola: $y = -2(x - 1)^2$ - Step 1: Identify the vertex form parameters: $a = -2$, $h = 1$, $k = 0$. - Step 2: The vertex is at $(1, 0)$. - Step 3: Since $a = -2$ is negative, the parabola opens downward and is narrower than $y = -x^2$ because $|a| = 2 > 1$. - Step 4: Plot the vertex at $(1, 0)$. - Step 5: Choose points around the vertex to plot: - For $x = 0$, $y = -2(0 - 1)^2 = -2(1) = -2$. - For $x = 2$, $y = -2(2 - 1)^2 = -2(1) = -2$. - Step 6: Plot points $(0, -2)$ and $(2, -2)$. - Step 7: Draw a smooth curve through these points forming a downward opening parabola. 6. Next, graph the fourth parabola: $y = -3(x + 2)^2 + 9$ - Step 1: Identify parameters: $a = -3$, $h = -2$, $k = 9$. - Step 2: The vertex is at $(-2, 9)$. - Step 3: Since $a = -3$ is negative, the parabola opens downward and is narrower than $y = -x^2$ because $|a| = 3 > 1$. - Step 4: Plot the vertex at $(-2, 9)$. - Step 5: Choose points around the vertex: - For $x = -3$, $y = -3(-3 + 2)^2 + 9 = -3(1)^2 + 9 = 6$. - For $x = -1$, $y = -3(-1 + 2)^2 + 9 = -3(1)^2 + 9 = 6$. - Step 6: Plot points $(-3, 6)$ and $(-1, 6)$. - Step 7: Draw a smooth curve through these points forming a downward opening parabola. Final answers: - The first two parabolas are correctly graphed. - The third and fourth parabolas can be graphed by identifying vertex, direction, and plotting points as shown.