point-slope equation

(noun)

An equation of a line given a point $(x_{1}, y_{1})$ and a slope $m$: $ y-y_{1}=m(x-x_{1})$.

Examples of point-slope equation in the following topics:

  • Point-Slope Equations

    • The point-slope equation is another way to represent a line; only the slope and a single point are needed.
    • The point-slope equation is a way of describing the equation of a line.
    • Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
    • Then plug this point into the point-slope equation and solve for $y$ to get:
    • Plug this point and the calculated slope into the point-slope equation to get:

  • 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 $(2,4)$
    • Therefore, the new equation has a slope of $-4$, through the point $(2,4)$.  

  • Slope

    • Slope describes the direction and steepness of a line, and can be calculated given two points on the line.
    • Putting the equation of a line into this form gives you the slope ($m$) of a line, and its $y$-intercept ($b$).
    • Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between any two distinct points on a line.
    • Starting with the point on the left, $(0, -3)$, sketch a right triangle, going from the first point to the second point, $(5, 1)$.
    • Calculate the slope of a line using "rise over run" and identify the role of slope in a linear equation

  • Solving Systems Graphically

    • A simple way to solve a system of equations is to look for the intersecting point or points of the equations.
    • This point is considered to be the solution of the system of equations.
    • To do this, you need to convert the equations to slope-intercept form, or $y=mx+b$, where m = slope and b = y-intercept.
    • Once you have converted the equations into slope-intercept form, you can graph the equations.
    • To determine the solutions of the set of equations, identify the points of intersection between the graphed equations.

  • Difference Quotients

    • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
    • Its "input value" is its argument, usually a point ("P") expressible on a graph.
    • The difference quotient is the average slope of a function between two points.
    • To show how the above equation can be written as the average slope, put it into more familiar terms.
    • The difference quotient can be used to calculate the average slope (here, represented by a straight line) between two points P and Q.

  • Slope-Intercept Equations

    • If an equation is not in slope-intercept form, solve for $y$ and rewrite the equation.
    • 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 $b=-2$.
    • The value of the slope dictates where to place the next point.
    • Thus we arrive at the point $(2,-5)$ on the line.

  • 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 $\displaystyle -\frac{A}{B}$.
    • where $(x_1, y_1)$ is any point on the line.
    • 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, $x-x_1$).

  • Linear Equations in Standard Form

    • 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.
    • The zero is the point $(5, 0)$.
    • Note that the $y$-intercept and slope can also be calculated using the coefficients and constant of the standard form equation.
    • If $B$ is non-zero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\frac{C}{B}$, and the slope of the line is $-\frac{A}{B}$.

  • 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.  
    • In the linear function graphs below, 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.
    • 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, $c$.  
    • For example, the graph of the equation $x=4$ includes the same input value of $4$ for all points on the line, but would have different output values, such as $(4,-2),(4,0),(4,1),(4,5),$ etcetera.

  • What is a Quadratic Function?

    • Quadratic equations are second order polynomials, and have the form $f(x)=ax^2+bx+c$.
    • Quadratic equations are different than linear functions in a few key ways.
    • Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope).
    • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.
    • where $h$ and $k$ are respectively the coordinates of the vertex, the point at which the function reaches either its maximum (if $a$ is negative) or minimum (if $a$ is positive).