Question

question #7 shania factored the rational equation in order to determine the vertical asymptote and point(s) of discontinuity. $f(x)=\frac{x^{2}-x - 56}{2x^{2}-98}$ shania’s work is shown. did the shania make a mistake? if so, in which step was the mistake first made? what was the mistake? $\frac{x^{2}-x - 56}{2x^{2}-98}$ step 1: $\frac{(x - 8)(x + 7)}{2(x^{2}-49)}$ step 2: $\frac{(x - 8)(x + 7)}{2(x + 7)(x - 7)}$ step 3: vertical asymptote(s): $x=-7 x = 7$ step 4: point of discontinuity : $(-7,\frac{15}{28})$ step 4; shania stated the points of discontinuity incorrectly. shania made no errors; all work is correct. step 3; shania stated the vertical asymptotes incorrectly. step 2; shania has the wrong factors for the denominator.

Explanation:

Step1: Factor numerator and denominator

For the numerator $x^{2}-x - 56$, we find two numbers that multiply to - 56 and add up to - 1. They are - 8 and 7, so $x^{2}-x - 56=(x - 8)(x + 7)$. For the denominator $2x^{2}-98$, we first factor out the common - factor 2: $2x^{2}-98 = 2(x^{2}-49)$, and then using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ with $a=x$ and $b = 7$, we get $2(x^{2}-49)=2(x + 7)(x - 7)$. So far, Shania's Step 1 and Step 2 are correct.

Step2: Determine vertical asymptotes

Vertical asymptotes occur at the values of $x$ that make the denominator of the simplified rational function equal to zero, after canceling out common factors. The simplified function is $\frac{x - 8}{2(x - 7)}$ (after canceling out the common factor $(x + 7)$). Setting the denominator $2(x - 7)=0$, we get $x = 7$. The value $x=-7$ is a removable discontinuity (a hole) since the factor $(x + 7)$ cancels out. So Shania made a mistake in Step 3 by including $x=-7$ as a vertical asymptote.

Step3: Determine point of discontinuity

The point of discontinuity (hole) occurs at the $x$ - value that makes the original common factor equal to zero. When $x=-7$, we substitute $x=-7$ into the simplified function $\frac{x - 8}{2(x - 7)}$. First, simplify the original function $\frac{(x - 8)(x + 7)}{2(x + 7)(x - 7)}=\frac{x - 8}{2(x - 7)}$. Substitute $x=-7$ into $\frac{x - 8}{2(x - 7)}$, we have $\frac{-7-8}{2(-7 - 7)}=\frac{-15}{2\times(-14)}=\frac{15}{28}$. Shania's Step 4 for the point of discontinuity is correct given the wrong vertical - asymptote determination in Step 3.

Answer:

C. Step 3; Shania stated the vertical asymptotes incorrectly.