1. Problem: Given $f(x) = -5x^2 + 10x - 5$, express in factored form, find vertex, zeros, axis of symmetry, direction of opening, domain, and range. Step 1: Factor the quadratic. Use the formula for factoring: $ax^2 + bx + c = a(x - r_1)(x - r_2)$ where $r_1, r_2$ are roots. Calculate discriminant: $\Delta = b^2 - 4ac = 10^2 - 4(-5)(-5) = 100 - 100 = 0$. Since $\Delta=0$, one repeated root: $x = \frac{-b}{2a} = \frac{-10}{2(-5)} = 1$. Factored form: $f(x) = -5(x - 1)^2$. Step 2: Find vertex. Vertex form is $f(x) = a(x - h)^2 + k$; here $h=1$, $k=0$. Vertex: $(1, 0)$. Step 3: Zeros. Zero at $x=1$ (repeated root). Step 4: Axis of symmetry. Line $x = 1$. Step 5: Direction of opening. Since $a = -5 < 0$, parabola opens downward. Step 6: Domain and range. Domain: all real numbers $(-\infty, \infty)$. Range: since vertex is max at $y=0$ and opens down, range is $(-\infty, 0]$. --- 2. Problem: For each function, state if max or min and method to find it. a) $f(x) = -2x^2 - 8x + 3$ - $a = -2 < 0$, so maximum value. - Method: Use vertex formula $x = -\frac{b}{2a} = -\frac{-8}{2(-2)} = 2$. - Calculate $f(2)$ for max value. b) $f(x) = 3(x - 1)(x + 5)$ - Expand: $3(x^2 + 4x - 5) = 3x^2 + 12x - 15$. - $a = 3 > 0$, so minimum value. - Method: Use vertex formula $x = -\frac{b}{2a} = -\frac{12}{2(3)} = -2$. - Calculate $f(-2)$ for min value. --- 4. Problem: Maximize area of rectangle with perimeter 2400 m. Step 1: Let length = $x$, width = $y$. Perimeter $P = 2(x + y) = 2400 \Rightarrow y = 1200 - x$. Step 2: Area $A = xy = x(1200 - x) = 1200x - x^2$. Step 3: Maximize $A$. Since $A = -x^2 + 1200x$, $a = -1 < 0$, max at vertex. Step 4: Vertex $x = -\frac{b}{2a} = -\frac{1200}{2(-1)} = 600$. Step 5: Width $y = 1200 - 600 = 600$. Step 6: Max area $A = 600 \times 600 = 360000$. --- 5. Problem: Find inverse of $f(x) = 2(x - 1)^2 - 3$. Step 1: Replace $f(x)$ with $y$: $y = 2(x - 1)^2 - 3$. Step 2: Swap $x$ and $y$: $x = 2(y - 1)^2 - 3$. Step 3: Solve for $y$. Add 3: $x + 3 = 2(y - 1)^2$. Divide by 2: $\frac{x + 3}{2} = (y - 1)^2$. Take square root: $y - 1 = \pm \sqrt{\frac{x + 3}{2}}$. Step 4: Solve for $y$: $y = 1 \pm \sqrt{\frac{x + 3}{2}}$. Step 5: Choose branch depending on domain/range. Inverse function: $f^{-1}(x) = 1 \pm \sqrt{\frac{x + 3}{2}}$. --- 6a. Problem: Simplify $(2 - \sqrt{8})(3 + \sqrt{2})$. Step 1: Expand using distributive property. $= 2 \times 3 + 2 \times \sqrt{2} - \sqrt{8} \times 3 - \sqrt{8} \times \sqrt{2}$ $= 6 + 2\sqrt{2} - 3\sqrt{8} - \sqrt{16}$ Step 2: Simplify radicals. $\sqrt{8} = 2\sqrt{2}$, $\sqrt{16} = 4$. $= 6 + 2\sqrt{2} - 3(2\sqrt{2}) - 4$ $= 6 + 2\sqrt{2} - 6\sqrt{2} - 4$ Step 3: Combine like terms. Constants: $6 - 4 = 2$. Radicals: $2\sqrt{2} - 6\sqrt{2} = -4\sqrt{2}$. Final: $2 - 4\sqrt{2}$. --- 6b. Problem: Simplify $(3 + \sqrt{5})(5 - \sqrt{10})$. Step 1: Expand. $= 3 \times 5 - 3 \times \sqrt{10} + \sqrt{5} \times 5 - \sqrt{5} \times \sqrt{10}$ $= 15 - 3\sqrt{10} + 5\sqrt{5} - \sqrt{50}$ Step 2: Simplify $\sqrt{50} = 5\sqrt{2}$. $= 15 - 3\sqrt{10} + 5\sqrt{5} - 5\sqrt{2}$. No like radicals to combine. Final: $15 - 3\sqrt{10} + 5\sqrt{5} - 5\sqrt{2}$. --- 6c. Explanation: Part (a) simplifies to fewer terms because radicals simplify and combine to like terms, while in part (b) the radicals are different and cannot be combined, resulting in more terms. --- 8. Problem: Does $g(x) = 6x - 5$ intersect $f(x) = 2x^2 - 3x + 2$? Find intersection points. Step 1: Set $g(x) = f(x)$. $6x - 5 = 2x^2 - 3x + 2$. Step 2: Rearrange to quadratic form. $0 = 2x^2 - 3x + 2 - 6x + 5 = 2x^2 - 9x + 7$. Step 3: Calculate discriminant. $\Delta = (-9)^2 - 4(2)(7) = 81 - 56 = 25 > 0$. Since $\Delta > 0$, two real roots, so two intersection points. Step 4: Find roots. $x = \frac{9 \pm \sqrt{25}}{2 \times 2} = \frac{9 \pm 5}{4}$. Roots: $x_1 = \frac{9 + 5}{4} = \frac{14}{4} = 3.5$, $x_2 = \frac{9 - 5}{4} = \frac{4}{4} = 1$. Step 5: Find $y$ values. $y_1 = g(3.5) = 6(3.5) - 5 = 21 - 5 = 16$. $y_2 = g(1) = 6(1) - 5 = 6 - 5 = 1$. Intersections: $(3.5, 16)$ and $(1, 1)$. --- 9. Problem: Find equation of parabola with vertex $(4, 3)$, zeros near $(1.5, 0)$ and $(6.5, 0)$, opening downward. Step 1: Use factored form: $f(x) = a(x - r_1)(x - r_2)$ with roots $r_1 = 1.5$, $r_2 = 6.5$. Step 2: Write $f(x) = a(x - 1.5)(x - 6.5)$. Step 3: Use vertex to find $a$. Vertex $x=4$, so $f(4) = a(4 - 1.5)(4 - 6.5) = a(2.5)(-2.5) = -6.25a$. Given $f(4) = 3$, so $3 = -6.25a \Rightarrow a = -\frac{3}{6.25} = -0.48$. Step 4: Equation: $f(x) = -0.48(x - 1.5)(x - 6.5)$. Step 5: Expand to standard form. $(x - 1.5)(x - 6.5) = x^2 - 8x + 9.75$. $f(x) = -0.48x^2 + 3.84x - 4.68$. Final equation: $f(x) = -0.48x^2 + 3.84x - 4.68$.