Find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the function. Use that information to sketch a graph. f(x) = \frac{x^2 – x – 30}{x^2 – 25} Enter the intercepts as points, (a,b). The x-intercept is The y-intercept is (0,1.2)
The Correct Answer and Explanation is:
Here is the solution based on the provided image.
Final Answer:
The x-intercept is (6, 0)
Explanation
To find the key features of the rational function f(x) = (x^2 – x – 30) / (x^2 – 25), we analyze its intercepts and asymptotes.
1. Factoring the Function
First, factor the numerator and the denominator to simplify the function and identify any holes.
- Numerator: x^2 – x – 30 = (x – 6)(x + 5)
- Denominator: x^2 – 25 = (x – 5)(x + 5)
The function in factored form is:
f(x) = [(x – 6)(x + 5)] / [(x – 5)(x + 5)]
We can see a common factor of (x + 5), which means there is a hole (a removable discontinuity) in the graph at x = -5.
2. Finding the Horizontal Intercept (x-intercept)
The x-intercepts occur where the function’s value is zero, which happens when the numerator is zero and the denominator is not. After simplifying the function by canceling the (x + 5) term, we are left with g(x) = (x – 6) / (x – 5).
Set the numerator of the simplified function to zero:
x – 6 = 0
x = 6
The x-intercept is the point where the graph crosses the x-axis, so the coordinates are (6, 0).
3. Finding the Vertical Intercept (y-intercept)
The y-intercept occurs when x = 0. Substitute x = 0 into the original function:
f(0) = (0^2 – 0 – 30) / (0^2 – 25) = -30 / -25 = 6/5 = 1.2
The y-intercept is at the point (0, 1.2), which confirms the value given in the problem.
4. Finding Asymptotes
- Vertical Asymptote: This occurs where the denominator of the simplified function is zero.
x – 5 = 0 => x = 5
There is a vertical asymptote at x = 5. - Horizontal Asymptote: We compare the degrees of the polynomials in the numerator (degree 2) and the denominator (degree 2). Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
y = 1 / 1 = 1
There is a horizontal asymptote at y = 1.
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