Question
1. For each of the following, determine: i) the y-intercept iii) the x-intercepts (zeros) (ii) the x-intercepts the vertex (a) y=9x^2-90x ii) the equation in factored form (factor it) iv) the axis of symmetry vi) sketch of the graph showing the above features (b) y=9x^2-81
Solution
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Regina
Professional · Tutor for 6 years
Answer
**(a) y = 9x² - 90x**i) **y-intercept:** The y-intercept occurs when x = 0. y = 9(0)² - 90(0) = 0 The y-intercept is 0.ii) **Equation in factored form:** y = 9x(x - 10)iii) **x-intercepts (zeros):** The x-intercepts occur when y = 0. 9x(x - 10) = 0 So, x = 0 or x = 10 The x-intercepts are 0 and 10.iv) **Axis of symmetry:** The axis of symmetry is the vertical line passing through the x-coordinate of the vertex. Since the x-intercepts are 0 and 10, the x-coordinate of the vertex is the average of these values: (0 + 10)/2 = 5. The axis of symmetry is x = 5.v) **Vertex:** The x-coordinate of the vertex is 5 (as calculated above). Substitute x = 5 into the equation to find the y-coordinate: y = 9(5)² - 90(5) = 225 - 450 = -225 The vertex is (5, -225).vi) **Sketch of the graph:**The graph is a parabola that opens upwards (since the coefficient of x² is positive). It passes through the points (0,0) and (10,0). The vertex is at (5, -225). The axis of symmetry is the vertical line x=5. (A hand-drawn sketch would show these features.)**(b) y = 9x² - 81**i) **y-intercept:** The y-intercept occurs when x = 0. y = 9(0)² - 81 = -81 The y-intercept is -81.ii) **Equation in factored form:** This is a difference of squares: y = (3x - 9)(3x + 9) or y = 9(x - 3)(x + 3)iii) **x-intercepts (zeros):** The x-intercepts occur when y = 0. 9(x - 3)(x + 3) = 0 So, x = 3 or x = -3 The x-intercepts are 3 and -3.iv) **Axis of symmetry:** The x-coordinate of the vertex is the average of the x-intercepts: (3 + (-3))/2 = 0. The axis of symmetry is x = 0.v) **Vertex:** The x-coordinate of the vertex is 0. Substitute x = 0 into the equation to find the y-coordinate: y = 9(0)² - 81 = -81 The vertex is (0, -81).vi) **Sketch of the graph:**The graph is a parabola that opens upwards. It passes through the points (3,0) and (-3,0). The vertex is at (0,-81), which is also the y-intercept. The axis of symmetry is the y-axis (x=0). (A hand-drawn sketch would show these features.)