Question

ip: graphing linear equations
for the following problems, identify the slope, y-intercept, and write the corresponding equation.
reminder: \\(y = mx + b\\)

a.
slope = _______
y-intercept = _______
equation: _______

b.
slope = _______
y-intercept = _______
equation: _______

c.
slope = _______
y-intercept = _______
equation: _______

Explanation:

Identify y-intercepts from the graphs

For each graph, we locate where the line crosses the vertical y-axis to find the y-intercept \(b\).

• For graph a, the line crosses the y-axis at \(2\) units above the origin. Thus, the y-intercept is \(2\) (or as a point, \((0, 2)\)).
• For graph b, the line crosses the y-axis at \(1\) unit above the origin. Thus, the y-intercept is \(1\) (or as a point, \((0, 1)\)).
• For graph c, the line is horizontal and lies exactly on the x-axis, crossing the y-axis at the origin \(0\). Thus, the y-intercept is \(0\) (or as a point, \((0, 0)\)).

Calculate slopes from the graphs

We find the slope \(m = \frac{\text{rise}}{\text{run}}\) by identifying two points with integer coordinates on each line.

• For graph a, the line passes through \((0, 2)\) and \((2, 3)\).

$$m = \frac{3 - 2}{2 - 0} = \frac{1}{2}$$

• For graph b, the line passes through \((0, 1)\) and \((1, -1)\).

$$m = \frac{-1 - 1}{1 - 0} = -2$$

• For graph c, the line is horizontal. Any two points, such as \((0, 0)\) and \((1, 0)\), give:

$$m = \frac{0 - 0}{1 - 0} = 0$$

Write the corresponding linear equations

Using the slope-intercept form \(y = mx + b\), we substitute the slope \(m\) and y-intercept \(b\) for each line.

• For graph a:

Using Linear Equation Solving principles:

$$y = \frac{1}{2}x + 2$$

• For graph b:

$$y = -2x + 1$$

• For graph c:

$$y = 0x + 0 \implies y = 0$$

Answer:

Question a

Slope = <blank>\(\frac{1}{2}\)</blank>

y-intercept = <blank>\(2\)</blank>

equation: <blank>\(y = \frac{1}{2}x + 2\)</blank>

Question b

Slope = <blank>\(-2\)</blank>

y-intercept = <blank>\(1\)</blank>

equation: <blank>\(y = -2x + 1\)</blank>

Question c

Slope = <blank>\(0\)</blank>

y-intercept = <blank>\(0\)</blank>

equation: <blank>\(y = 0\)</blank>