Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{94-2 x^{2}}{3 x^{2}-12} $$

Step-by-Step Solution

Answer

x-intercepts: \(x=\sqrt{47}, -\sqrt{47}\); y-intercept: \(y=-\frac{47}{6}\).

Step 1: Find the x-intercepts

To find the x-intercepts of the function, set the numerator equal to zero and solve for \( x \). The function is \( f(x)=\frac{94-2x^2}{3x^2-12} \). Thus, solve \( 94-2x^2=0 \).Rearrange to \( 2x^2=94 \).Divide by 2: \( x^2 = 47 \).Taking the square root gives \( x = \pm \sqrt{47} \). Hence, the x-intercepts are \( x = \sqrt{47} \) and \( x = -\sqrt{47} \).

Step 2: Simplify To Find the y-intercept

To find the y-intercept, evaluate \( f(x) \) at \( x = 0 \). Substitute \( x = 0 \) into the function:\( f(0) = \frac{94 - 2(0)^2}{3(0)^2 - 12} = \frac{94}{-12} \).Simplify the fraction: \( f(0) = \frac{47}{-6} = -\frac{47}{6} \).Thus, the y-intercept is \( y = -\frac{47}{6} \).

Step 3: Finalize Intercepts

To summarize, the coordinates of the x-intercepts are \( (\sqrt{47}, 0) \) and \( (-\sqrt{47}, 0) \). The y-intercept is at \( (0, -\frac{47}{6}) \). These points can be used to sketch the graph and understand where the function crosses the axes.

Key Concepts

x-intercepts

The x-intercepts of a function are points where the graph crosses or touches the x-axis. This occurs when the value of the function is zero.
To find the x-intercepts of an algebraic or rational function, you need to set the numerator equal to zero and solve for the variable, usually represented by \( x \).
For the function given in the example, \( f(x)=\frac{94-2x^2}{3x^2-12} \), this process involved the following steps:

• Set the numerator, \( 94 - 2x^2 \), equal to zero: \( 94 - 2x^2 = 0 \).
• This leads to the equation \( 2x^2 = 94 \).
• By dividing both sides by 2, we get \( x^2 = 47 \).
• Taking the square root of both sides, we find \( x = \pm \sqrt{47} \).

Thus, the x-intercepts are \( x = \sqrt{47} \) and \( x = -\sqrt{47} \). These represent the points \( (\sqrt{47}, 0) \) and \( (-\sqrt{47}, 0) \) on the coordinate plane.

y-intercepts

Y-intercepts occur where the graph crosses the y-axis. At this point, the value of \( x \) is always zero, thus to find the y-intercept of any function, simply substitute \( x = 0 \) into the function and solve for \( y \).
In rational functions like our example \( f(x)=\frac{94-2x^2}{3x^2-12} \), this is straightforward:

• Substitute \( x = 0 \) into the function, giving \( f(0) = \frac{94-2(0)^2}{3(0)^2-12} \).
• Simplify this expression to \( f(0) = \frac{94}{-12} \).
• Reducing the fraction results in \( f(0) = -\frac{47}{6} \).

Thus, the y-intercept is \( y = -\frac{47}{6} \), represented by the point \( (0, -\frac{47}{6}) \), showing where the graph intersects the y-axis.

rational functions

Rational functions are ratios of two polynomials. They take the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).
These functions can behave differently at their intercepts, vertical asymptotes, and horizontal asymptotes.

• The intercepts, as discussed previously, occur where the graph crosses the axes.
• Vertical asymptotes occur where the denominator \( q(x) \) equals zero, provided it doesn't simultaneously zero the numerator.
• Horizontal asymptotes depend on the degrees of \( p(x) \) and \( q(x) \).
  • If the degree of \( p(x) \) is less than that of \( q(x) \), the x-axis \( y = 0 \) is the asymptote.
  • If the degree of \( p(x) \) equals that of \( q(x) \), the horizontal asymptote is at \( y = \frac{text{leading coefficient of } p(x)}{text{leading coefficient of } q(x)} \).
  • Finally, if the degree of \( p(x) \) is greater than that of \( q(x) \), there is no horizontal asymptote, but possibly a slant asymptote.

This foundational understanding of rational functions is essential for analyzing and graphing these functions accurately.