An expression is shown below: f(x) = -16x2 + 8x + 3 Part A: What are the x-intercepts of the graph of the f(x)? Show your work. (2 points) Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points) Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

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An expression is shown below: f(x) = -16x2 + 8x + 3 Part A: What are the x-intercepts of the graph of the f(x)? Show your work. (2 points) Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points) Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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A) To get x-intercepts, set f(x) (or y) to zero and solve for x. Solve -16x^2 + 24x + 16 = 0 divide throughout by -8 2x^2 - 3x - 2 = 0 solve for x. 2x^2 - 3x - 2 = 0 2x^2 - 4x + x - 2 = 0 2x(x - 2) + 1(x - 2) = 0 (2x+1)(x-2) = 0 x = -0.5 or x = 2 The x-intercepts are: -0.5 and 2. B) Is the vertex of the graph of f(x) going to be a maximum or minimum? f(x) = -16x^2 + 24x + 16 The "a" or the coefficient of x^2 is -16. Since it is a negative, it is an inverted parabola that opens downward. Therefore, the vertex will be a maximum. To find the vertex, complete the square of -16x^2 + 24x + 16 The vertex form of this equation will look like: f(x) = a(x-h)^2 + k where (h,k) is the vertex. Complete the square of -16x^2 + 24x + 16 Make the coefficient of x^2 1 by factoring out -16 -16(x^2 - 24/16x - 1) = -16(x^2 - 3/2x - 1). To complete the square: Divide the coefficient of x by 2: -3/2 / 2 = -3/4. This will go inside the parenthesis to be squared and you will have to subtract the square of it: -16(x^2 - 3/2x - 1) = -16{ (x-3/4)^2 - (-3/4)^2 - 1 } = -16{ (x-3/4)^2 - 9/16 - 1 } = -16{ (x-3/4)^2 - 25/16 } = f(x) = -16(x-3/4)^2 + 25 compare this to f(x) = a(x-h)^2 + k where (h,k) is the vertex and you can see that h = 3/4 and k = 25. So the vertex is (3/4, 25) or (0.75, 25) C) f(x) = -16x^2 + 24x + 16 We know the graph is a parabola. We also know it is an inverted parabola because of the negative coefficient of x^2. We know the curve crosses the x-axis at x = -0.5 and x = 2 as they are the x-intercepts found in part A). From Part B) we know that the vertex of the parabola is at (0.75, 25). We can also set x = 0 and find the y-intercept: -16(0)^2 + 24(0) + 16 = 16. So the curve crosses the y-axis at y = 16 We have enough information to plot the graph.

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