Question

charlie graphs the equation $y = -\frac{1}{2}x + 2$. select all statements about charlies graph that are true. the y-intercept of the line is 2. the slope of the line is $\frac{1}{2}$. the line does not pass through the origin. the graph is a straight line. the line passes through the point $(0, -2)$.

Explanation:

The equation of the line is given in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the equation \(y=-\frac{1}{2}x + 2\):

Step 1: Analyze the y - intercept

In the slope - intercept form \(y=mx + b\), when \(x = 0\), \(y=b\). Here, \(b = 2\), so the y - intercept is 2. So the statement "The y - intercept of the line is 2" is true.

Step 2: Analyze the slope

In the equation \(y=mx + b\), \(m\) is the slope. For \(y =-\frac{1}{2}x+2\), \(m=-\frac{1}{2}\), not \(\frac{1}{2}\). So the statement "The slope of the line is \(\frac{1}{2}\)" is false.

Step 3: Check if the line passes through the origin

The origin is \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the equation: \(0=-\frac{1}{2}(0)+2\), which simplifies to \(0 = 2\), a false statement. So the line does not pass through the origin, and the statement "The line does not pass through the origin" is true.

Step 4: Determine the type of graph

The equation \(y=-\frac{1}{2}x + 2\) is a linear equation (in the form \(y=mx + b\)), and the graph of a linear equation is a straight line. So the statement "The graph is a straight line" is true.

Step 5: Check if the line passes through \((0,-2)\)

Substitute \(x = 0\) into the equation \(y=-\frac{1}{2}(0)+2=2\). When \(x = 0\), \(y = 2\), not \(-2\). So the line does not pass through \((0,-2)\), and the statement "The line passes through the point \((0,-2)\)" is false.

Answer:

The true statements are:

• The y - intercept of the line is 2.
• The line does not pass through the origin.
• The graph is a straight line.

In boxed form (for the checkboxes, we can represent the true ones as selected):

• \(\boxed{\text{The y - intercept of the line is 2.}}\)
• \(\boxed{\text{The line does not pass through the origin.}}\)
• \(\boxed{\text{The graph is a straight line.}}\)