Question

graph the equation $y + 4 = -\frac{1}{2}(x + 2)$.

Explanation:

Step1: Identify the form of the equation

The given equation is \( y + 4 = -\frac{1}{2}(x + 2) \), which is in the point - slope form \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-2,-4) \) and the slope \( m =-\frac{1}{2}\).

Step2: Analyze the slope and the point

• The slope \( m=-\frac{1}{2} \) is negative, so the line should be decreasing (going down from left to right).
• The line passes through the point \( (-2,-4) \).

Step3: Evaluate the options (by checking the slope and the point)

• For a line with a negative slope, we can eliminate the option with a positive - sloped line (the second graph).
• Now, check the point \( (-2,-4) \). Let's also rewrite the equation in slope - intercept form (\( y=mx + b \)) to find the y - intercept.
• Start with \( y + 4=-\frac{1}{2}(x + 2) \).
• Distribute the \( -\frac{1}{2} \): \( y + 4=-\frac{1}{2}x-1 \).
• Subtract 4 from both sides: \( y=-\frac{1}{2}x-1 - 4=-\frac{1}{2}x-5 \). The y - intercept is \( b=-5 \).
• The third graph (the one with the line going down and passing near \( x = 0,y=-5 \) and passing through \( (-2,-4) \)) matches the slope (negative) and the y - intercept (\( y=-5 \)) and the point \( (-2,-4) \). The first graph's y - intercept is not - 5, and the second graph has a positive slope.

Answer:

The third graph (the one with the line having a negative slope, passing through \( (-2,-4) \) and with y - intercept - 5)