Question

(06.03)
which system of equations is represented by the graph?
options:
○ a ( y = x + 4 ) ( y=\frac{x + 4}{x + 2} )
○ b ( y = x - 4 ) ( y=\frac{x + 4}{x + 2} )
○ c ( y = x + 4 ) ( y=\frac{x - 4}{x + 2} )
○ d ( y = x - 4 ) ( y=\frac{x - 4}{x + 2} )

Explanation:

Step1: Analyze the linear equation

The red line is a linear function. Let's use the two points \((4, 0)\) and \((-1, -5)\) to find its equation. The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-5 - 0}{-1 - 4}=\frac{-5}{-5} = 1\). Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((4,0)\), we get \(y-0 = 1\times(x - 4)\), so \(y=x - 4\).

Step2: Analyze the rational equation

The blue curve is a rational function. The vertical asymptote: For a rational function \(y=\frac{f(x)}{g(x)}\), the vertical asymptote occurs where \(g(x)=0\). From the options, the denominator is \(x + 2\), so the vertical asymptote is \(x=-2\) (which matches the graph as the curve has a vertical asymptote at \(x = - 2\)). The \(x\) - intercept of the rational function: Set \(y = 0\), then \(\frac{f(x)}{x + 2}=0\), so \(f(x)=0\). From the graph, the rational function has an \(x\) - intercept at \(x = 4\). Let's check the numerator:

• For option a: Numerator \(x + 4\), when \(x=-4\), \(y = 0\) (not \(x = 4\)), so a is wrong.
• For option b: Numerator \(x + 4\), when \(x=-4\), \(y = 0\) (not \(x = 4\)), so b is wrong.
• For option c: Numerator \(x - 4\), when \(x = 4\), \(y = 0\), but the linear equation is \(y=x + 4\) (we found the linear equation is \(y=x - 4\) from step 1), so c is wrong.
• For option d: Numerator \(x - 4\), when \(x = 4\), \(y = 0\) (matches the \(x\) - intercept of the rational function), and the linear equation is \(y=x - 4\) (matches the linear equation we found in step 1). Also, the vertical asymptote of \(y=\frac{x - 4}{x + 2}\) is \(x=-2\) (matches the graph).

Answer:

d. \(y = x-4\), \(y=\frac{x - 4}{x + 2}\)