1. **Problem Statement:** Find the x-intercepts, axis of symmetry, and vertex for each quadratic equation given in factored form. 2. **Formula and Rules:** - X-intercepts: Set $y=0$ and solve for $x$ from each factor. - Axis of Symmetry: The midpoint between the two x-intercepts, calculated as $\frac{x_1 + x_2}{2}$. - Vertex: Substitute the axis of symmetry $x$ value into the original equation to find $y$. --- ### a) $y = (x - 2)(x - 10)$ - X-intercepts: $x=2$, $x=10$ - Axis of Symmetry: $\frac{2 + 10}{2} = 6$ - Vertex: $y = (6 - 2)(6 - 10) = 4 \times (-4) = -16$ - Vertex coordinate: $(6, -16)$ --- ### b) $y = (x + 3)(x - 5)$ - X-intercepts: $x=-3$, $x=5$ - Axis of Symmetry: $\frac{-3 + 5}{2} = 1$ - Vertex: $y = (1 + 3)(1 - 5) = 4 \times (-4) = -16$ - Vertex coordinate: $(1, -16)$ --- ### c) $y = 2(x + 5)(x + 1)$ - X-intercepts: $x=-5$, $x=-1$ - Axis of Symmetry: $\frac{-5 + (-1)}{2} = -3$ - Vertex: $y = 2(-3 + 5)(-3 + 1) = 2 \times 2 \times (-2) = -8$ - Vertex coordinate: $(-3, -8)$ --- ### d) $y = -1(x - 2)(x - 4)$ - X-intercepts: $x=2$, $x=4$ - Axis of Symmetry: $\frac{2 + 4}{2} = 3$ - Vertex: $y = -1(3 - 2)(3 - 4) = -1 \times 1 \times (-1) = 1$ - Vertex coordinate: $(3, 1)$ --- ### e) $y = (x + 2)(x - 5)$ - X-intercepts: $x=-2$, $x=5$ - Axis of Symmetry: $\frac{-2 + 5}{2} = 1.5$ - Vertex: $y = (1.5 + 2)(1.5 - 5) = 3.5 \times (-3.5) = -12.25$ - Vertex coordinate: $(1.5, -12.25)$ --- ### f) $y = 3(x + 7)(x + 2)$ - X-intercepts: $x=-7$, $x=-2$ - Axis of Symmetry: $\frac{-7 + (-2)}{2} = -4.5$ - Vertex: $y = 3(-4.5 + 7)(-4.5 + 2) = 3 \times 2.5 \times (-2.5) = -18.75$ - Vertex coordinate: $(-4.5, -18.75)$ --- ### g) $y = -2(x - 1)(x + 7)$ - X-intercepts: $x=1$, $x=-7$ - Axis of Symmetry: $\frac{1 + (-7)}{2} = -3$ - Vertex: $y = -2(-3 - 1)(-3 + 7) = -2 \times (-4) \times 4 = 32$ - Vertex coordinate: $(-3, 32)$ --- ### h) $y = (x - 1.4)(x - 5.6)$ - X-intercepts: $x=1.4$, $x=5.6$ - Axis of Symmetry: $\frac{1.4 + 5.6}{2} = 3.5$ - Vertex: $y = (3.5 - 1.4)(3.5 - 5.6) = 2.1 \times (-2.1) = -4.41$ - Vertex coordinate: $(3.5, -4.41)$