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 $-1$.  

• 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 $5$.

• 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 $-\frac{A}{B}$.

• 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.

• Graphing Quadratic Equations In Standard Form

  • A quadratic function is a polynomial function of the form $y=ax^2+bx+c$.
  • For example, consider the parabola $y=2x^2-4x+4 $ shown below.
  • More specifically, it is the point where the parabola intercepts the y-axis.
  • The point $(0,c)$ is the $y$ intercept of the parabola.
  • Note that the parabola above has $c=4$ and it intercepts the $y$-axis at the point $(0,4).$

• Direct Variation

  • Revisiting the example with toothbrushes and dollars, we can define the x axis as number of toothbrushes and the y axis as number of dollars.
  • For example, doubling y would result in the doubling of x.
  • Graph of direct variation with the linear equation y=0.8x.
  • The line y=kx is an example of direct variation between variables x and y.
  • For all points on the line, y/x=k.

• Solving Systems Graphically

  • 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.

• 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 $A$ is nonzero, then the x-intercept, or the x-coordinate of the point where the graph crosses the x-axis (where $y$ is zero), is $\displaystyle \frac{C}{A}$.
  • 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 $\displaystyle -\frac{A}{B}$.
  • Graph sample of linear equations, using the y=mx+b format, as seen by $y=-x+5$(red) and $y=\frac{1}{2}x +2$ (blue).

• 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 $(5,0)$.