a) The factor form of the polynomial is f(x) = 4 · (x - 3) · (x + 5 / 4).
b) The values of the x-intercepts of the graph of f(x) are represented by the roots found in part a).
c) The function has a minimum and grows in the + y direction.
d) Marking the two x-intercepts, second, generating a line perpendicular to the x-axis that passes the midpoint of the line segment generated by the two intercepts and estimate the location of the vertex and, finally, creating a parabola with a minimum and growing in the + y direction by matching the points.
In this problem we have a quadratic function in standard form, that is, an equation of the form f(x) = a · x² + b · x + c, where a, b and c are real coefficients. a) First, we need to complete the square and find the roots to determine the factorized form:
4 · x² - 7 · x - 15 = 0
4 · [x² - (7 / 4) · x - (15 / 4)] = 0
x² - (7 / 4) · x - (15 / 4) = 0
x² - 2 · (7 / 8) · x - (15 / 4) + (289 / 64) = 289 / 64
x² - 2 · (7 / 8) · x + 49 / 64 = 289 / 64
(x - 7 / 8)² = 289 / 64
x - 7 / 8 = ± 17 / 8
x = 7 / 8 ± 17 / 8
Then, the factor form of the polynomial is f(x) = 4 · (x - 3) · (x + 5 / 4).
b) The values of the x-intercepts of the graph of f(x) are represented by the roots found in part a).
c) The end behavior of the graph is inferred from the lead coefficient. A positive coeffcient , which is the case of the function, indicates that vertex of the polynomial is a minimum and parabola grows in the + y direction.
d) In accordance with algebra, we can generate a quadratic equation by knowing three distinct points on Cartesian plane.
First, mark the two x-intercepts, second, generate a line perpendicular to the x-axis that passes the midpoint of the line segment generated by the two intercepts and, finally, generate a parabola with a minimum and growing in the + y direction.
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