y-intercept

Examples of y-intercept in the following topics:

• Slope-Intercept Equations

  • Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.  
  • 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 $b=-2$.
  • We begin by plotting the $y$-intercept $b=-2$, whose coordinates are $(0,-2)$.  
  • The slope is $2$, and the $y$-intercept is $-1$.  

• Slope and Y-Intercept of a Linear Equation

  • For the linear equation y = a + bx, b = slope and a = y-intercept.
  • From algebra recall that the slope is a number that describes the steepness of a line and the y-intercept is
  • the y coordinate of the point ( 0,a ) where the line crosses the y-axis.
  • What is the y-intercept and what is the slope?
  • The y-intercept is 25 (a = 25).

• Slope and Intercept

  • Using the common convention that the horizontal axis represents a variable $x$ and the vertical axis represents a variable $y$, a $y$-intercept is a point where the graph of a function or relation intersects with the $y$-axis of the coordinate system.
  • If the curve in question is given as $y=f(x)$, the $y$-coordinate of the $y$-intercept is found by calculating $f(0)$.
  • Functions which are undefined at $x=0$ have no $y$-intercept.
  • Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one $y$-intercept.
  • Because functions associate $x$ values to no more than one $y$ value as part of their definition, they can have at most one $y$-intercept.

• 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.
  • 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 $5$.

• Slope and Intercept

  • In the regression line equation the constant $m$ is the slope of the line and $b$ is the $y$-intercept.
  • Here, by convention, $x$ and $y$ are the variables of interest in our data, with $y$ the unknown or dependent variable and $x$ the known or independent variable.
  • The constant $$$m$ is slope of the line and $b$ is the $y$-intercept -- the value where the line cross the $y$ axis.
  • Linear regression is an approach to modeling the relationship between a scalar dependent variable $y$ and one or more explanatory (independent) variables denoted $X$.
  • An equation where y is the dependent variable, x is the independent variable, m is the slope, and b is the intercept.

• Linear Equations in Standard Form

  • For example, consider an equation in slope-intercept form: $y = -12x +5$.
  • Recall that a zero is a point at which a function's value will be equal to zero ($y=0$), and is the $x$-intercept of the function.
  • We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form.
  • 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 $-\frac{A}{B}$.

• The Equation of a Line

  • The intercept of the fitted line is such that it passes through the center of mass $(x, y)$ of the data points.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • Where $m$ (slope) and $b$ (intercept) designate constants.
  • 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.
  • Three lines — the red and blue lines have the same slope, while the red and green ones have same y-intercept.

• Parts of a Parabola

  • Parabolas also have an axis of symmetry, which is parallel to the y-axis.
  • The y-intercept is the point at which the parabola crosses the y-axis.
  • If there were, the curve would not be a function, as there would be two $y$ values for one $x$ value, at zero.
  • If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of $x$ at which $y=0$.
  • A parabola can have no x-intercepts, one x-intercept, or two x-intercepts.

• Point-Slope Equations

  • The point-slope form is ideal if you are given the slope and only one point, or if you are given two points and do not know what the $y$-intercept is.
  • Plug in the generic point into the equation $y=mx+b$.  
  • Then plug this point into the point-slope equation and solve for $y$ to get:
  • To switch this equation into slope-intercept form, solve the equation for $y$:
  • 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, $y=-4x+9$.

• Zeroes of Linear Functions

  • A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
  • An $x$-intercept, or zero, is a property of many functions.
  • Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.
  • To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
  • The blue line, $y=\frac{1}{2}x+2$, has a zero at $(-4,0)$; the red line, $y=-x+5$, has a zero at $(5,0)$.