I need help with all 3 of these!!!

Question

1.)Explain the procedure(but don’t create the graph) for graphing the quadratic function in vertex for f(x)=2(x-5)^2+3 using a pencil and paper. State the vertex, and how you found it.Give at least two other points on the graph, showing or explaining how you found them.

2.)Explain the procedure(but don’t create the graph) for graphing the function in factored form f(x)=(x-5)(x+3) using pencil and paper. Your explanation should show your understanding of the intercept graphing method. Make sure to state at least three points on the graph including the vertex.Show or explain how you found each point.

3.)Explain how the factored form of a quadratic equation and the functions zeros are connected. Write the factored form of f(x) and state the zeros of f(x). f(x)=x^2+10x-24

Answer 1

1.) The equation is in vertex form, so the vertex is at point (h, k) for the quadratic function f(x) = a(x-h)² + k. Therefore, in this case, the vertex is at point (5,3). You can find 2 other points simply by choosing 2 other values of x, and plugging them into the function to find their corresponding y-values, giving you two other points on the parabola. (Ex.: (6,5), (7,11))

2.) The factored form gives you the x-intercepts of the function. a and b are x-intercepts of the function f(x) = (x-a)(x-b), therefore the x-intercepts here are 5 and -3. Now, the x-coordinate of the vertex will always be halfway between the two x-intercepts, or in this case (5-3)/2 = 1. Plug this value into the function to get (1-5)(1+3) = -16, therefore the vertex is at (1,-16). Therefore, 3 points on the graph are (5,0), (-3,0), and (1,-16).

3.) The zeros of a quadratic function are the x-intercepts, which are directly given by the factored form of the function. We can factor the given function as:

x² + 10x - 24 = (x+12)(x-2) = 0

Since we are multiplying 2 terms with a result of 0, at least one of the two terms must be 0, therefore we have:

x + 12 = 0

x - 2 = 0

Therefore, the zeros of the function are x = -12 and x = 2.