Question
Quadraties Test 2 Quiz Combo 1. For each of the following, determine: i) the y-intercept iii) the x-intercepts (zeros) 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 (c) y=9x^2+23x-6 (d) y=9x^2+18x-27
Solution
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Ryleigh
Elite · Tutor for 8 years
Answer
**1(a) *y* = 9*x*² - 90*x**** **i) y-intercept:** Set *x* = 0. *y* = 9(0)² - 90(0) = 0. The y-intercept is 0.* **ii) Factored form:** *y* = 9*x*(*x* - 10)* **iii) x-intercepts (zeros):** Set *y* = 0. 9*x*(*x* - 10) = 0. This gives *x* = 0 and *x* = 10. The x-intercepts are 0 and 10.* **iv) Axis of symmetry:** The axis of symmetry is the average of the x-intercepts: *x* = (0 + 10)/2 = 5* **v) Vertex:** The x-coordinate of the vertex is the same as the axis of symmetry: *x* = 5. Substitute this into the original equation (or the factored form) to find the y-coordinate: *y* = 9(5)² - 90(5) = 225 - 450 = -225. The vertex is (5, -225).* **vi) Sketch:** The parabola opens upwards (since the coefficient of *x*² is positive). It passes through the points (0,0), (10,0), and has a vertex at (5,-225).**1(b) *y* = 9*x*² - 81*** **i) y-intercept:** Set *x* = 0. *y* = 9(0)² - 81 = -81. The y-intercept is -81.* **ii) Factored form:** *y* = 9(*x*² - 9) = 9(*x* - 3)(*x* + 3) (Difference of squares)* **iii) x-intercepts (zeros):** Set *y* = 0. 9(*x* - 3)(*x* + 3) = 0. This gives *x* = 3 and *x* = -3. The x-intercepts are 3 and -3.* **iv) Axis of symmetry:** *x* = (-3 + 3)/2 = 0* **v) Vertex:** The x-coordinate of the vertex is 0. *y* = 9(0)² - 81 = -81. The vertex is (0, -81).* **vi) Sketch:** The parabola opens upwards and passes through (-3,0), (3,0) with a vertex at (0,-81).**1(c) *y* = 9*x*² + 23*x* - 6*** **i) y-intercept:** Set *x* = 0. *y* = -6.* **ii) Factored form:** This quadratic doesn't factor nicely using integers. We can use the quadratic formula to find the roots and then write it in factored form.* **iii) x-intercepts (zeros):** Use the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a. x = [-23 ± sqrt(23² - 4(9)(-6))] / 18. x = [-23 ± sqrt(529 + 216)] / 18. x = [-23 ± sqrt(745)] / 18. So, x ≈ 0.24 and x ≈ -2.79.* **ii) Factored Form (using the approximate roots):** y ≈ 9(x - 0.24)(x + 2.79) (Note: This is an approximation due to the rounded roots.)* **iv) Axis of symmetry:** *x* ≈ (0.24 - 2.79)/2 ≈ -1.275* **v) Vertex:** *x* ≈ -1.275. Substitute into the original equation: *y* ≈ 9(-1.275)² + 23(-1.275) - 6 ≈ -21.5. Vertex ≈ (-1.275, -21.5)* **vi) Sketch:** Parabola opens upwards, passing through the approximate x-intercepts and with the calculated vertex.**1(d) *y* = 9*x*² + 18*x* - 27*** **i) y-intercept:** -27* **ii) Factored form:** *y* = 9(*x*² + 2*x* - 3) = 9(*x* + 3)(*x* - 1)* **iii) x-intercepts (zeros):** -3 and 1* **iv) Axis of symmetry:** *x* = (-3 + 1)/2 = -1* **v) Vertex:** *x* = -1. *y* = 9(-1)² + 18(-1) - 27 = 9 - 18 - 27 = -36. Vertex is (-1, -36).* **vi) Sketch:** Parabola opens upwards, passing through (-3,0), (1,0) with vertex at (-1,-36).