for problems 5-6, determine each of the following: (a) the direction of the parabola. (b) the x-intercepts. (c) the axis of symmetry. (d) the vertex. (e) the intervals of increase/decrease. (f) the graph. 5) y = -(x - 1)(x + 3) 6) y = 2(x - 2)(x - 4)

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### Problem 5: $ y = -(x - 1)(x + 3) $ #### (a) Direction of the parabola # Answer: Downward # Brief Explanations: The leading coefficient (after expanding) is $-1$ (negative), so the parabola opens downward. #### (b) $ x $-intercepts # Answer: $ x = 1 $ and $ x = -3 $ # Brief Explanations: Set $ y = 0 $: $ 0 = -(x - 1)(x + 3) $. Solve $ (x - 1) = 0 $ or $ (x + 3) = 0 $, giving $ x = 1 $ and $ x = -3 $. #### (c) Axis of symmetry # Answer: $ x = -1 $ # Brief Explanations: The axis of symmetry is the midpoint of the $ x $-intercepts: $ x = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1 $. #### (d) Vertex # Answer: $ (-1, 4) $ # Explanation: ## Step 1: Find $ x $-coordinate of vertex The axis of symmetry is $ x = -1 $, so the $ x $-coordinate of the vertex is $ -1 $. ## Step 2: Find $ y $-coordinate of vertex Substitute $ x = -1 $ into $ y = -(x - 1)(x + 3) $: $ y = -(-1 - 1)(-1 + 3) = -(-2)(2) = -(-4) = 4 $. #### (e) Intervals of increase/decrease # Answer: Increases on $ (-\infty, -1) $; Decreases on $ (-1, \infty) $ # Brief Explanations: Since the parabola opens downward, it increases to the left of the vertex ($ x < -1 $) and decreases to the right of the vertex ($ x > -1 $). #### (f) Graph (Description): A parabola opening downward with $ x $-intercepts at $ (1, 0) $ and $ (-3, 0) $, vertex at $ (-1, 4) $, and axis of symmetry $ x = -1 $. --- ### Problem 6: $ y = 2(x - 2)(x - 4) $ #### (a) Direction of the parabola # Answer: Upward # Brief Explanations: The leading coefficient (after expanding) is $ 2 $ (positive), so the parabola opens upward. #### (b) $ x $-intercepts # Answer: $ x = 2 $ and $ x = 4 $ # Brief Explanations: Set $ y = 0 $: $ 0 = 2(x - 2)(x - 4) $. Solve $ (x - 2) = 0 $ or $ (x - 4) = 0 $, giving $ x = 2 $ and $ x = 4 $. #### (c) Axis of symmetry # Answer: $ x = 3 $ # Brief Explanations: The axis of symmetry is the midpoint of the $ x $-intercepts: $ x = \frac{2 + 4}{2} = \frac{6}{2} = 3 $. #### (d) Vertex # Answer: $ (3, -2) $ # Explanation: ## Step 1: Find $ x $-coordinate of vertex The axis of symmetry is $ x = 3 $, so the $ x $-coordinate of the vertex is $ 3 $. ## Step 2: Find $ y $-coordinate of vertex Substitute $ x = 3 $ into $ y = 2(x - 2)(x - 4) $: $ y = 2(3 - 2)(3 - 4) = 2(1)(-1) = -2 $. #### (e) Intervals of increase/decrease # Answer: Decreases on $ (-\infty, 3) $; Increases on $ (3, \infty) $ # Brief Explanations: Since the parabola opens upward, it decreases to the left of the vertex ($ x < 3 $) and increases to the right of the vertex ($ x > 3 $). #### (f) Graph (Description): A parabola opening upward with $ x $-intercepts at $ (2, 0) $ and $ (4, 0) $, vertex at $ (3, -2) $, and axis of symmetry $ x = 3 $.

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Explanation:

Problem 5: \( y = -(x - 1)(x + 3) \)

(a) Direction of the parabola

(b) \( x \)-intercepts

(c) Axis of symmetry

(d) Vertex

Answer: