To solve an equation by finding x-intercepts, Cynthia graphed the function y=x^3-5x+|2x-3|. Give several possibilities for the original equation.
Thanks
first of all change to
|2x-3| = y - x^3 + 5x
then
a) 2x - 3 = y - x^3 + 5x or
b) -2x + 3 = y - x^3 + 5x
solve each one for y
To find the x-intercepts of a function, we need to determine the values of x for which the function crosses the x-axis (i.e., where y is equal to zero).
In this case, Cynthia graphed the function y = x^3 - 5x + |2x - 3|. To find the x-intercepts, we set y equal to zero and solve for x.
1. Setting y = 0, we have 0 = x^3 - 5x + |2x - 3|.
- One possibility for the original equation is: x^3 - 5x + |2x - 3| = 0.
To simplify the absolute value expression, we consider two cases:
Case 1: (2x - 3) ≥ 0, i.e., when 2x - 3 ≥ 0.
- When 2x - 3 ≥ 0, the absolute value operation becomes unnecessary, so we keep it as it is.
- In this case, the equation can be written as: x^3 - 5x + (2x - 3) = 0.
Case 2: (2x - 3) < 0, i.e., when 2x - 3 < 0.
- When 2x - 3 < 0, the absolute value expression becomes the negation of its content.
- In this case, the equation can be written as: x^3 - 5x - (2x - 3) = 0.
Hence, several possible original equations are:
- x^3 - 5x + |2x - 3| = 0.
- x^3 - 5x + (2x - 3) = 0.
- x^3 - 5x - (2x - 3) = 0.
These equations represent different possibilities, taking into account the absolute value expression.