Final answer:
The equation of the parabola is y = -0.2(x+5)^2 + 8. When x=15, the value of y is -72. The average rate of change between x=-5 and x=15 is -4.
Step-by-step explanation:
The given information allows us to write the equation of the parabola in vertex form, which is y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola. Since the vertex is given as (-5, 8), we can substitute these values in to get y = a(x+5)^2 + 8. We also know the parabola passes through the point (5, -12), which allows us to find the value of 'a' by plugging in the values of x and y: -12 = a(5+5)^2 + 8. Solving for 'a' gives us -20 = 100a ⇖ a = -0.2. Therefore, the equation of the parabola is y = -0.2(x+5)^2 + 8.
To find the value of y when x = 15, we substitute 15 for x in the parabola equation: y = -0.2(15+5)^2 + 8. This simplifies to y = -0.2(20)^2 + 8, which gives us y = -72The average rate of change of a function on an interval [a, b] is found by taking the difference in y-values divided by the difference in x-values: (y2 - y1) / (x2 - x1). For the interval x = -5 to x = 15, we use the y-values at those x-values, which are 8 (given for x = -5 at the vertex) and -72 (calculated above). The average rate of change is (-72 - 8) / (15 - (-5)) = -80 / 20 = -4. Therefore, the average rate of change is -4The equation of a parabola that passes through a given point and has a given vertex can be determined using the standard form of the equation for a parabola: y = a(x - h)^2 + k, where (h, k) is the vertex. Since the line of symmetry is parallel to the y-axis, the equation can be simplified to y = a(x - h)^2 + k.Using the given information, the vertex is (-5, 8). Substituting this into the equation, we get: y = a(x + 5)^2 + 8.Next, we substitute the point (5, -12) into the equation and solve for a. This gives us the equation: -12 = a(5 + 5)^2 + 8. Solving for a, we find that a = -2/5.Therefore, the equation of the parabola is: y = -2/5(x + 5)^2 + 8When x = 15, the value of y can be calculated by substituting x = 15 into the equation: y = -2/5(15 + 5)^2 + 8. Solving fory, we get y = -13..