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The slope and $y$-intercept of a line help to describe it. In this lesson you will look at equations such as $y=3x+2$ and their graphs, and learn about each graph’s shape, slope, and $y$-intercept.
The slope of a line measures how steep the line is. It is the amount of change in the height of a line as you go 1 unit to the right. If the height of a line decreases as you move to the right, its slope is negative.
Use the slider in the lower left portion of your screen to change the value of $m$ in the equation $y=mx$ and complete the table below. Use the grid to the left to determine each slope. The equation for $y$ is shown below the grid.
The $y$-intercept of a line is the point where it crosses the $y$-axis. Use the slider to change the value of $b$ in the equation $y=x+b$ and complete the table below.
So far you have learned that the slope of the line $y=3x$ is 3 and that the $y$-intercept of the line $y=x+2$ is $(0,2)$, but what about the slope and $y$-intercept of $y=3x+2$? Is the graph of $y=3x+2$ a straight line? In this section we will answer these questions.
Each row of this table gives an equation for a line. Find the $y$-intercept of that line algebraically, by setting $x$ to $0$ in its equation.
You can now see that:
The $y$-intercept of the line $y=mx+b$ is the point $(0, b)$.
Each row of this table gives an equation for a line. Find the slopes of these lines algebraically, by subtracting their $y$-values when $x=0$ and $x=1$.
As you can see:
The slope of the line $y=mx+b$ is $m$.
As you have learned, the numbers $m$ and $b$ in the equation $y=mx+b$ give the slope and $y$-intercept of the line that is the graph of that equation. Because of this:
$y=mx+b$ is called the slope-intercept form of the equation for a line.
Use the sliders to change the values of $m$ and $b$ and answer the following questions.
Use the sliders to change $m$ and $b$ and answer the following questions.
Use the sliders to change the values of $m$ and $b$ and complete the table below.
The points $(0,1)$ and $(2,-3)$ are shown on the grid. Use the sliders to answer the following questions.