1. **State the problem:** We are given two quadratic functions: - $f(x) = (x - 2)^2 + 5$ - $g(x) = -2(x + 3)^2 + 8$ We need to analyze each parabola by answering questions about their properties. --- ### For $f(x) = (x - 2)^2 + 5$: 1. **Opens up or down?** Since the coefficient of the squared term is positive (1), the parabola opens **upwards**. 2. **Axis of symmetry:** The axis of symmetry is the vertical line through the vertex. For $f(x) = (x - h)^2 + k$, axis is $x = h$. Here, $h = 2$, so axis of symmetry is: $$x = 2$$ 3. **Vertex:** The vertex is at $(h, k)$, so: $$(2, 5)$$ 4. **y-intercept:** Set $x=0$: $$f(0) = (0 - 2)^2 + 5 = 4 + 5 = 9$$ So y-intercept is: $$(0, 9)$$ 5. **Zeros:** Set $f(x) = 0$: $$(x - 2)^2 + 5 = 0 \\ (x - 2)^2 = -5$$ Since square of a real number cannot be negative, there are **no real zeros**. 6. **Max or min?** Since it opens upwards, vertex is a **minimum** point. Minimum value is $5$ at $x=2$. 7. **Domain $D$:** All real numbers: $$(-\infty, +\infty)$$ 8. **Range $R$:** Since minimum is 5 and parabola opens up: $$[5, +\infty)$$ 9. **Standard form:** Expand: $$f(x) = (x - 2)^2 + 5 = (x^2 - 4x + 4) + 5 = x^2 - 4x + 9$$ --- ### For $g(x) = -2(x + 3)^2 + 8$: 1. **Opens up or down?** Coefficient of squared term is $-2$ (negative), so it opens **downwards**. 2. **Axis of symmetry:** For $g(x) = a(x - h)^2 + k$, axis is $x = h$. Here, $h = -3$, so: $$x = -3$$ 3. **Vertex:** At $(h, k) = (-3, 8)$ 4. **y-intercept:** Set $x=0$: $$g(0) = -2(0 + 3)^2 + 8 = -2(9) + 8 = -18 + 8 = -10$$ So y-intercept is: $$(0, -10)$$ 5. **Zeros:** Set $g(x) = 0$: $$-2(x + 3)^2 + 8 = 0 \\ -2(x + 3)^2 = -8 \\ (x + 3)^2 = 4 \\ x + 3 = \pm 2$$ So zeros are: $$x = -3 + 2 = -1$$ $$x = -3 - 2 = -5$$ --- **Final answers:** $f(x)$: - Opens up - Axis: $x=2$ - Vertex: $(2,5)$ - y-intercept: $(0,9)$ - Zeros: none - Min at $y=5$ - Domain: $(-\infty,+\infty)$ - Range: $[5,+\infty)$ - Std form: $x^2 - 4x + 9$ $g(x)$: - Opens down - Axis: $x=-3$ - Vertex: $(-3,8)$ - y-intercept: $(0,-10)$ - Zeros: $x=-1, x=-5$