Question

eustace used the following graphed rational function to verify he had determined the correct x and y intercepts. determine if eustace made any errors in his work and where the error(s) occurred. $f(x)=\frac{2x^{2}-2x - 12}{x^{2}-4x + 3}$
eustaces work
step 1 y - intercept $y=\frac{2(0)^{2}-2(0)-12}{(0)^{2}-4(0)+3}=-4$ y - int: $(0,4)$
step 2 factor $f(x)=\frac{2(x + 2)(x - 3)}{(x - 1)(x - 3)}$
step 3 reduce $f(x)=\frac{2(x + 2)}{x - 1}$
step 4 x - intercept x - int: $(-2,0)$
eustace did not make any errors. all work is correct as well as the values of the x and y intercepts.
eustace made an error in step 2. he factored the rational function incorrectly.
eustace made an error in step 1. the y - intercept should be y - int: $(0,-4)$.
eustace made an error in step 4. the x - intercept should be x - int: $(2,0)$

Explanation:

Step1: Check y - intercept calculation

The formula for the y - intercept is to set \(x = 0\) in \(y=\frac{2x^{2}-2x - 12}{x^{2}-4x + 3}\). When \(x = 0\), \(y=\frac{2(0)^{2}-2(0)-12}{(0)^{2}-4(0)+3}=\frac{- 12}{3}=-4\). The y - intercept should be \((0,-4)\), but Eustace wrote \((0,4)\) in step 1.

Step2: Check factoring

Factor the numerator \(2x^{2}-2x - 12=2(x^{2}-x - 6)=2(x + 2)(x - 3)\) and the denominator \(x^{2}-4x + 3=(x - 1)(x - 3)\). Eustace's factoring in step 2 is correct.

Step3: Check reduction

After factoring, \(\frac{2(x + 2)(x - 3)}{(x - 1)(x - 3)}=\frac{2(x + 2)}{x - 1}\) (for \(x
eq3\)). Eustace's reduction in step 3 is correct.

Step4: Check x - intercept

To find the x - intercept, set \(y = 0\) in \(y=\frac{2(x + 2)}{x - 1}\). Then \(2(x + 2)=0\), which gives \(x=-2\). The x - intercept \((-2,0)\) in step 4 is correct.

Answer:

Eustace made an error in step 1. The y - intercept should be y - int: \((0,-4)\)