Question

which graph matches the equation $y + 6 = \frac{3}{4}(x + 4)$?

first graph with points (-6, -4), (0, -0.5), (6, 5)

second graph with points (-4, -6), (0, -9), (4, -12)

third graph with points (-4, -6), (0, -3), (4, 0)

fourth graph with points (-6, -4), (0, -8.5), (2, -10)

Explanation:

Step1: Identify the form of the equation

The given equation is \( y + 6=\frac{3}{4}(x + 4) \), which is in the point - slope form \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(- 4,-6) \) and the slope \( m = \frac{3}{4} \). So the line should pass through the point \( (-4,-6) \) and have a positive slope (since \( m=\frac{3}{4}>0 \)).

Step2: Analyze the slope

The slope \( m=\frac{3}{4} \) means that for a run of 4 units (change in \( x \)), the rise is 3 units (change in \( y \)). Let's check the points. Starting from \( (-4,-6) \), if we move 4 units to the right (increase \( x \) by 4: \( x=-4 + 4=0 \)), we should move 3 units up (increase \( y \) by 3: \( y=-6+3=-3 \)). So the line should also pass through \( (0,-3) \).

Step3: Analyze each graph

• First graph: The points given are \( (-6,-4) \), \( (0, - 0.5) \), \( (6,5) \). The slope between \( (-6,-4) \) and \( (0,-0.5) \) is \( \frac{-0.5+4}{0 + 6}=\frac{3.5}{6}=\frac{7}{12}

eq\frac{3}{4} \), so this is not the correct graph.

• Second graph: The slope between \( (-4,-6) \), \( (0,-9) \) and \( (4,-12) \) is \( \frac{-9 + 6}{0+4}=\frac{-3}{4} \) (negative slope), which does not match \( m = \frac{3}{4} \), so this is not correct.
• Third graph: The points are \( (-4,-6) \), \( (0,-3) \), \( (4,0) \). The slope between \( (-4,-6) \) and \( (0,-3) \) is \( \frac{-3 + 6}{0 + 4}=\frac{3}{4} \), and the slope between \( (0,-3) \) and \( (4,0) \) is \( \frac{0 + 3}{4-0}=\frac{3}{4} \), which matches the slope of the given equation. Also, it passes through \( (-4,-6) \) and \( (0,-3) \) as we calculated.
• Fourth graph: The slope between \( (-6,-4) \), \( (0,-8.5) \) and \( (2,-10) \) is negative, so this is not correct.

Answer:

The third graph (the one with points \((-4,-6)\), \((0,-3)\), \((4,0)\))