The parabola is g(x) = ax2 + q.

Step 1: Analyze the given information and graph. The parabola is g(x) = ax^2 + q. The straight line is f(x) = mx + c. R and S(2; 0) are the x-intercepts of g. T(0; 8) is the y-intercept of g. Graph f passes through R and T. Since g(x) = ax^2 + q is a parabola with its axis of symmetry at x=0 (the y-axis), and S(2; 0) is an x-intercept, the other x-intercept R must be symmetric to S with respect to the y-axis. Therefore, R is (-2; 0). T(0; 8) is the y-intercept of g. For a parabola of the form ax^2+q, the y-intercept is also the vertex. Since the parabola has x-intercepts and the vertex is above the x-axis, it opens downwards. 5.1: Write down the range of g. The vertex of g(x) = ax^2 + q is at (0, q). From the graph, T(0, 8) is the vertex. Since the parabola opens downwards, the maximum value of g(x) is the y-coordinate of the vertex. The maximum value is y=8. The range of g is all real numbers less than or equal to 8. Range of g: y 8 5.2: Write down the x-coordinate of R. As established in Step 1, due to the symmetry of g(x) = ax^2 + q about the y-axis, and S(2; 0) being an x-intercept, R must be the symmetric x-intercept. x-coordinate of R: -2 5.3: Calculate the values of a and q. From T(0; 8) being the y-intercept of g(x) = ax^2 + q, we substitute x=0 and g(x)=8: 8 = a(0)^2 + q 8 = q So, q=8. The equation for g(x) is now g(x) = ax^2 + 8. We use an x-intercept, for example S(2; 0), to find a. Substitute x=2 and g(x)=0: 0 = a(2)^2 + 8 0 = 4a + 8 4a = -8 a = (-8)/(4) a = -2 The values are a = -2 and q = 8. a = -2, q = 8 5.4: Determine the equation of f. The function f(x) = mx + c is a straight line that passes through R(-2; 0) and T(0; 8). First, calculate the slope m: m = (y_2 - y_1)/(x_2 - x_1) = (8 - 0)/(0 - (-2)) = (8)/(2) = 4 Next, find the y-intercept c. Since T(0; 8) is a point on the line and its x-coordinate is 0, the y-coordinate 8 is the y-intercept. So, c = 8. The equation of f is f(x) = 4x + 8. f(x) = 4x + 8 5.5: Use the graphs to determine the value(s) of x for which: 5.5.1: f(x) = g(x) This condition means finding the x-coordinates where the graphs of f and g intersect. From the graph, the intersection points are R and T. R has x-coordinate -2. T has x-coordinate 0. x = -2 or x = 0 5.5.2: x · g(x) 0 We know g(x) = -2x^2 + 8. The x-intercepts of g(x) are x=-2 and x=2. The parabola g(x) is above the x-axis (i.e., g(x) 0) for x [-2, 2]. The parabola g(x) is below the x-axis (i.e., g(x) 0) for x (-, -2] [2, ). We need x · g(x) 0. This occurs when: 1. x 0 AND g(x) 0: This means x (-, 0] and x [-2, 2]. The intersection is x [-2, 0]. 2. x 0 AND g(x) 0: This means x [0, ) and x (-, -2] [2, ). The intersection is x [2, ). Combining these two cases, the solution is x [-2, 0] [2, ). x [-2, 0] [2, ) 5.6: The graph h is obtained when g is reflected along the line y=0. Write down the equation of h in the form h(x) = px^2 + k. Reflecting a function y = g(x) along the line y=0 (the x-axis) means that the new y-values are the negative of the original y-values. So, h(x) = -g(x). We found g(x) = -2x^2 + 8. h(x) = -(-2x^2 + 8) h(x) = 2x^2 - 8 This is in the form h(x) = px^2 + k, where p=2 and k=-8. h(x) = 2x^2 - 8 Send me the next one 📸