1. **State the problem:** Given the function $f(x) = (x - 2)(2x - 12)$, find the y-intercept, x-intercepts, line of symmetry, vertex, value at $x = -1$, and intervals where the function is increasing or decreasing. 2. **Expand the function:** $$f(x) = (x - 2)(2x - 12) = 2x^2 - 12x - 4x + 24 = 2x^2 - 16x + 24$$ 3. **Find the y-intercept:** The y-intercept occurs when $x=0$. $$f(0) = 2(0)^2 - 16(0) + 24 = 24$$ So, the y-intercept is $(0, 24)$. 4. **Find the x-intercepts:** Set $f(x) = 0$: $$(x - 2)(2x - 12) = 0$$ This gives two equations: $$x - 2 = 0 \Rightarrow x = 2$$ $$2x - 12 = 0 \Rightarrow 2x = 12 \Rightarrow x = 6$$ So, the x-intercepts are $(2, 0)$ and $(6, 0)$. 5. **Equation for the line of symmetry:** The line of symmetry for a quadratic $ax^2 + bx + c$ is given by: $$x = -\frac{b}{2a}$$ Here, $a=2$, $b=-16$: $$x = -\frac{-16}{2 \times 2} = \frac{16}{4} = 4$$ So, the line of symmetry is $x = 4$. 6. **Find the vertex:** The vertex lies on the line of symmetry. Find $f(4)$: $$f(4) = 2(4)^2 - 16(4) + 24 = 2(16) - 64 + 24 = 32 - 64 + 24 = -8$$ So, the vertex is at $(4, -8)$. 7. **Find $f(-1)$:** $$f(-1) = 2(-1)^2 - 16(-1) + 24 = 2(1) + 16 + 24 = 42$$ 8. **Determine intervals of increase and decrease:** Since $a=2 > 0$, the parabola opens upward. - It decreases on $(-\infty, 4)$ - It increases on $(4, \infty)$ **Final answers:** - a) y-intercept: $(0, 24)$ - b) x-intercepts: $(2, 0)$ and $(6, 0)$ - c) Line of symmetry: $x = 4$ - d) Vertex: $(4, -8)$ - e) $f(-1) = 42$ - f) Increasing on $(4, \infty)$ - g) Decreasing on $(-\infty, 4)$