To find the intercepts and asymptotes of the function , we will follow these steps:
y-intercept: To find the y-intercept, we substitute into the function and solve for .
x-intercepts: To find the x-intercepts, we set and solve for .
Vertical asymptotes: Vertical asymptotes occur when the denominator of the function is equal to zero.
Horizontal asymptote: To find the horizontal asymptote, we compare the degrees of the numerator and denominator of the function.
Let’s start by finding the y-intercept.
Evaluate the expression by following steps:
step0: Evaluate:
step1: Substitute:
step2: Calculate:
step3: Calculate:
step4: Multiply:
step5: Multiply:
step6: Multiply:
step7: Multiply:
step8: Remove 0:
step9: Remove 0:
step10: Rewrite the fraction:
The y-intercept of the function is at the point .
Next, let’s find the x-intercepts by setting and solving for .
Solve the equation by following steps:
step0: Solve for :
step1: Find the domain:
step2: Cross multiply:
step3: Simplify the equation:
step4: Factor the expression:
step5: Separate into possible cases:
step6: Solve the equation:
step7: Check if the solution is in the defined range:
step8: Find the intersection:
step9: Rewrite:
The x-intercepts of the function are at the points and .
Next, let’s find the vertical asymptotes by setting the denominator of the function equal to zero and solving for .
Solve the quadratic equation by following steps:
step0: Solve by factoring:
step1: Factor the expression:
step2: Separate into possible cases:
step3: Solve the equation:
step4: Rewrite:
The vertical asymptotes of the function occur at and .
Finally, let’s determine the horizontal asymptote by comparing the degrees of the numerator and denominator of the function. Since the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
The horizontal asymptote is .
Therefore, the function has:
A y-intercept at the point
X-intercepts at the points and
Vertical asymptotes at and
A horizontal asymptote at
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To find the intercepts, asymptotes, and characteristics of the function :
The -intercept occurs when :
So, the -intercept is at the point .
The -intercepts occur when the numerator equals zero . Factoring gives:
Thus, the -intercepts are at the points and .
For vertical asymptotes, set the denominator equal to zero:
Using the quadratic formula:
This gives and . So, the vertical asymptotes are at and .
For the horizontal asymptote, compare the leading coefficients of the numerator and denominator: