slope-intercept form

Examples of slope-intercept form in the following topics:

• Slope-Intercept Equations

  • One of the most common representations for a line is with the slope-intercept form.
  • Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.  
  • Simply substitute the values into the slope-intercept form to obtain:
  • Let's write the equation $3x+2y=-4$ in slope-intercept form and identify the slope and $y$-intercept.
  • Now that the equation is in slope-intercept form, we see that the slope $m=-\frac{3}{2}$, and the $y$-intercept $b=-2$.

• Point-Slope Equations

  • Example: Write the equation of a line in point-slope form, given a point $(2,1)$ and slope $-4$, and convert to slope-intercept form
  • To switch this equation into slope-intercept form, solve the equation for $y$:
  • Example: Write the equation of a line in point-slope form, given point $(-3,6)$ and point $(1,2)$, and convert to slope-intercept form
  • Graph of the line $y-1=-4(x-2)$, through the point $(2,1)$ with slope of $-4$, as well as the slope-intercept form, $y=-4x+9$.
  • Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation

• Solving Systems Graphically

  • Before successfully solving a system graphically, one must understand how to graph equations written in standard form, or $Ax+By=C$.
  • To do this, you need to convert the equations to slope-intercept form, or $y=mx+b$, where m = slope and b = y-intercept.
  • The best way to convert an equation to slope-intercept form is to first isolate the y variable and then divide the right side by B, as shown below.
  • Now $\displaystyle -\frac{A}{B}$ is the slope m, and $\displaystyle \frac{C}{B}$ is the y-intercept b.
  • Once you have converted the equations into slope-intercept form, you can graph the equations.

• Parallel and Perpendicular Lines

  • Recall that the slope-intercept form of an equation is: $y=mx+b$ and the point-slope form of an equation is: $y-y_{1}=m(x-x_{1})$, both contain information about the slope, namely the constant $m$.
  • Example:  Write an equation of the line (in slope-intercept form) that is parallel to the line $y=-2x+4$ and passes through the point $(-1,1)$
  • Start with the equation for slope-intercept form and then substitute the values for the slope and the point, and solve for $b$: $y=mx+b$.  
  • Example:  Write an equation of the line (in slope-intercept form) that is perpendicular to the line $y=\frac{1}{4}x-3$ and passes through the point $(2,4)$
  • Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.  

• Linear Equations in Standard Form

  • A linear equation written in standard form makes it easy to calculate the zero, or $x$-intercept, of the equation.
  • For example, consider an equation in slope-intercept form: $y = -12x +5$.
  • We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form.
  • However, the zero, or $x$-intercept of a linear equation can easily be found by putting it into standard form.
  • Note that the $y$-intercept and slope can also be calculated using the coefficients and constant of the standard form equation.

• Direct and Inverse Variation

  • Notice that this is a linear equation in slope-intercept form, where the $y$-intercept $b$ is equal to $0$.
  • It is impossible to put it in slope-intercept form.
  • Thus, an inverse relationship cannot be represented by a line with constant slope.
  • Relate the concept of slope to the concepts of direct and inverse variation

• What is a Linear Function?

  • Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
  • For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.  
  • Vertical lines have an undefined slope, and cannot be represented in the form $y=mx+b$, but instead as an equation of the form $x=c$ for a constant $c$, because the vertical line intersects a value on the $x$-axis, $c$.  
  • Horizontal lines have a slope of zero and is represented by the form, $y=b$, where $b$ is the $y$-intercept.  
  • The blue line has a positive slope of $\frac{1}{2}$ and a $y$-intercept of $-3$; the red line has a negative slope of $-1$ and a $y$-intercept of $5$.

• Linear and Quadratic Equations

  • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
  • These equations are often referred to as the "equations of the straight line. " In what follows, $x$, $y$, $t$, and $\theta$ are variables, $m$ is the slope, and $b$ is the y-intercept.
  • If $B$ is nonzero, then the y-intercept, or the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\displaystyle \frac{C}{B}$, and the slope of the line is $\displaystyle -\frac{A}{B}$.
  • Vertical lines, having undefined slopes, cannot be represented by this form.
  • The point-slope form expresses the fact that the difference in the y-coordinate between two points on a line (that is, $y-y_1$) is proportional to the difference in the x-coordinate (that is, $x-x_1$).

• Fitting a Curve

  • With linear regression, a line in slope-intercept form, $y=mx+b$ is found that "best fits" the data.
  • To find the slope of the line of best fit, calculate in the following steps:
  • To find the $y$-intercept ($b$) , calculate using the following steps:
  • First, find the slope $(m)$ and $y$-intercept $(b)$ that best approximate this data, using the equations from the prior section:
  • The denominator is $92-\frac{1}{8}(20)^{2}=92-50=42$ and the slope is the quotient of the numerator and denominator: $\frac{23.25}{42}\approx0.554.$

• Slope

  • Slope is often denoted by the letter $m$.
  • Recall the slop-intercept form of a line, $y = mx + b$.
  • Putting the equation of a line into this form gives you the slope ($m$) of a line, and its $y$-intercept ($b$).
  • In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.
  • We can see the slope is decreasing, so be sure to look for a negative slope.