We want to find the x- and y-intercepts of the graph of the given equation. Let's begin with the x-intercept.
Finding the x-intercept
Think of the point where the graph of an equation crosses the x-axis. This is the x-intercept. The y-value of that ( x, y) coordinate pair is 0, so to find the x-intercept of the equation we should substitute 0 for y and solve for x.
We received a polynomial equation. Note that each term contains x. This allows us to factor it out.
2x^3-10x^2-x=0 ⇒ x(2x^2-10x-1)=0
The left-hand side is a product of two factors. Therefore, we can apply the Zero Product Property to solve the equation.
From Equation (I) we found that one solution is x=0. To find other solutions, we will solve Equation (II). Note that this is a quadratic equation. Thus, we will use the Quadratic Formula.
ax^2+bx+c=0 ⇔ x=- b±sqrt(b^2-4ac)/2a
To do so, we first need to identify a, b, and c.
2x^2-10x-1=0 ⇔ 2x^2+( - 10)x+( - 1)=0
We see that a= 2, b= - 10, and c= - 1. Let's substitute these values into the formula and solve for x.
Using the Quadratic Formula, we found that another two solutions of the equation are 5±3sqrt(3)2.
Solutions
x=0, x=5-3sqrt(3)/2, x=5+3sqrt(3)/2
The equation has 3 solutions, so the graph of the given equation has 3 x-intercepts: (0,0), ( 5-3sqrt(3)2,0), and ( 5+3sqrt(3)2,0).
Finding the y-intercept
Let's use the same concept to find the y-intercept. Consider the point where the graph of the equation crosses the y-axis. The x-value of the ( x, y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give us the y-intercept.
Now we can use the definition of a logarithm to rewrite it in exponential form.
Definition:& n=log_b m &&⇔ b^n= m
Equation:& 2=log_3 (x-1) &&⇔ 3^2= x-1
Once we have isolated x, we can solve the equation.
