Question

directions: at the top right comer of your paper, please include the following:

• your full legal first & last name(s) capitalize the first letter of both/all
• todays date no weekend, holidays, or non-school days

instructions: answer the following questions in the spaces provided. this assignment is not graded, you will get full credit as
long as you answer everything.

1. what is slope intercept form?
2. what is slope?
3. what is y - intercept?
4. is the line y = 2x - 3 increasing or

decreasing? how do you know?

1. what do we need to remember about

graphing < or > in one dimension and 2
dimensions?

1. what do we need to remember about

graphing ≤ or ≥ in one dimension and 2
dimensions?

1. what is the equation of the graph?
2. what is the equation of the graph?

Explanation:

Step1: Slope - intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Slope

The slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). It represents the rate of change of \(y\) with respect to \(x\).

Step3: \(y\) - intercept

The \(y\) - intercept \(b\) is the value of \(y\) when \(x = 0\). In the equation \(y=mx + b\), it is the point \((0,b)\) where the line crosses the \(y\) - axis.

Step4: Increasing or decreasing line

For the line \(y = 2x-3\), since the slope \(m = 2>0\), the line is increasing. When the slope is positive, as \(x\) increases, \(y\) also increases.

Step5: Graphing \(<\) or \(>\)

• One - dimension (\(x\) - axis): For \(x>a\) (or \(x < a\)), we use an open circle at \(x = a\) and shade the region to the right (for \(x>a\)) or to the left (for \(x < a\)).
• Two - dimension (\(x - y\) plane): For \(y>mx + b\) (or \(ymx + b\)) or below (for \(y

Step6: Graphing \(\leq\) or \(\geq\)

• One - dimension (\(x\) - axis): For \(x\geq a\) (or \(x\leq a\)), we use a closed circle at \(x = a\) and shade the region to the right (for \(x\geq a\)) or to the left (for \(x\leq a\)).
• Two - dimension (\(x - y\) plane): For \(y\geq mx + b\) (or \(y\leq mx + b\)), we draw a solid line (because the points on the line are included in the solution set) and shade the region above (for \(y\geq mx + b\)) or below (for \(y\leq mx + b\)) the line.

Step7: Equation of the first graph

The graph has an open circle at \(x = 1\) and is shaded to the right. The equation is \(x>1\).

Step8: Equation of the second graph

First, find the equation of the boundary line. The line passes through \((0,1)\) and \((- 2,-3)\). The slope \(m=\frac{-3 - 1}{-2-0}=\frac{-4}{-2}=2\). Using the \(y\) - intercept \(b = 1\), the equation of the line is \(y = 2x+1\). Since the line is solid and the region is shaded to the left of the line, the inequality is \(y\leq2x + 1\).

Answer:

1. \(y=mx + b\)
2. \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
3. The value of \(y\) when \(x = 0\)
4. Increasing, because the slope \(m = 2>0\)
5. One - dimension: open circle; Two - dimension: dashed line
6. One - dimension: closed circle; Two - dimension: solid line
7. \(x>1\)
8. \(y\leq2x + 1\)