onsider the function y = −2x2 + 2x + 24.(a) Find its y-intercept.(b) Find its x-intercepts.(c) Sketch its graph, labelling the axes and intercepts clearly.(d) Hence solve the inequality −2x2 + 2x + 24 ≥ 0.

(a) The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0. So, to find the y-intercept, we substitute x = 0 into the equation:

y = -2(0)^2 + 2(0) + 24 y = 24

So, the y-intercept is 24.

(b) The x-intercepts of a function are the points where the graph of the function intersects the x-axis. This occurs when y = 0. So, to find the x-intercepts, we set y = 0 and solve for x:

0 = -2x^2 + 2x + 24 0 = x^2 - x - 12 (divide all terms by -2)

This is a quadratic equation, which can be factored:

0 = (x - 4)(x + 3)

Setting each factor equal to zero gives the solutions x = 4 and x = -3. So, the x-intercepts are 4 and -3.

(c) To sketch the graph, start by plotting the y-intercept (0, 24). Then plot the x-intercepts (4, 0) and (-3, 0). The graph is a downward-opening parabola because the coefficient of x^2 is negative. The vertex of the parabola is halfway between the x-intercepts, at x = 0.5, and can be found by substituting x = 0.5 into the equation to find y.

(d) To solve the inequality -2x^2 + 2x + 24 ≥ 0, we consider the x-intercepts found in part (b). The parabola is above the x-axis (i.e., y ≥ 0) between x = -3 and x = 4. So, the solution to the inequality is -3 ≤ x ≤ 4.