Question

1. accurately sketch the graph of the following functions (labeling important points) and determine the following characteristics for each function. 5 marks each

a. ( y = \frac{3x}{x^2 + 2x - 8} )
domain:
range:
x-intercept:
y-intercept:
equation of vertical asymptote(s):
equation of horizontal asymptote:
point(s) of discontinuity:
(grid for graphing is provided)

Explanation:

Step1: Find Domain

The function is \( y = \frac{3x}{x^2 + 2x - 8} \). First, factor the denominator: \( x^2 + 2x - 8=(x + 4)(x - 2) \). The denominator cannot be zero, so \( x
eq - 4 \) and \( x
eq2 \). Thus, domain is \( \{x\in\mathbb{R}|x
eq - 4,x
eq2\} \).

Step2: Find Range

To find the range, we can rewrite the function as \( y(x^2 + 2x - 8)=3x \), \( yx^2+(2y - 3)x-8y = 0 \). For real \( x \), the discriminant \( \Delta=(2y - 3)^2+32y^2\geq0 \). Expand: \( 4y^2-12y + 9+32y^2\geq0 \), \( 36y^2-12y + 9\geq0 \), \( 12y^2-4y + 3\geq0 \). The discriminant of this quadratic in \( y \) is \( (-4)^2-4\times12\times3=16 - 144=-128<0 \), so it is always positive. Thus, range is \( \mathbb{R} \).

Step3: X - Intercept

Set \( y = 0 \), then \( \frac{3x}{x^2 + 2x - 8}=0 \). So \( 3x = 0\Rightarrow x = 0 \). X - intercept is \( (0,0) \).

Step4: Y - Intercept

Set \( x = 0 \), \( y=\frac{0}{0 + 0 - 8}=0 \). Y - intercept is \( (0,0) \).

Step5: Vertical Asymptotes

Vertical asymptotes occur where denominator is zero (and numerator non - zero). Denominator zero at \( x=-4 \) and \( x = 2 \), numerator at these points: at \( x=-4 \), numerator \( 3\times(-4)=-12
eq0 \); at \( x = 2 \), numerator \( 3\times2 = 6
eq0 \). So vertical asymptotes: \( x=-4 \), \( x = 2 \).

Step6: Horizontal Asymptote

For rational function \( y=\frac{ax^n+\cdots}{bx^m+\cdots} \), if \( n

Step7: Points of Discontinuity

Discontinuities at \( x=-4 \) and \( x = 2 \) (where denominator is zero).

Answer:

• Domain: \( \boldsymbol{\{x\in\mathbb{R}:x

eq - 4,x
eq2\}} \)

• Range: \( \boldsymbol{\mathbb{R}} \)
• X - Intercept: \( \boldsymbol{(0,0)} \)
• Y - Intercept: \( \boldsymbol{(0,0)} \)
• Vertical Asymptotes: \( \boldsymbol{x=-4,x = 2} \)
• Horizontal Asymptote: \( \boldsymbol{y = 0} \)
• Points of Discontinuity: \( \boldsymbol{x=-4,x = 2} \)