linear function

Examples of linear function in the following topics:

• Zeroes of Linear Functions

  • A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
  • The graph of a linear function is a straight line.
  • Linear functions can have none, one, or infinitely many zeros.  
  • To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
  • The zero from solving the linear function above graphically must match solving the same function algebraically.

• What is a Linear Function?

  • Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
  • A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
  • It is linear: the exponent of the $x$ term is a one (first power), and it follows the definition of a function: for each input ($x$) there is exactly one output ($y$).  
  • The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.  
  • Identify what makes a function linear and the characteristics of a linear function

• What is a Quadratic Function?

  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • 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).
  • With a linear function, each input has an individual, unique output (assuming the output is not a constant).  
  • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

• Graphical Representations of Functions

  • Function notation, $f(x)$ is read as "$f$ of $x$" which means "the value of the function at $x$."  
  • The ordered pairs normally stated in linear equations as $(x,y)$, in function notation are now written as $(x,f(x))$.
  • The function is linear, since the highest degree in the function is a $1$.  
  • Only two points are required to graph a linear function.
  • The degree of the function is 3, therefore it is a cubic function.

• Linear and Quadratic Equations

  • Two kinds of equations are linear and quadratic.
  • Linear equations can have one or more variables.
  • Linear equations do not include exponents.
  • Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as $xy$, $x^2$, $y^{\frac{1}{3}}$, and $\sin{x}$ are nonlinear.
  • Plots of the real-valued quadratic function $ax^2 + bx + c$, varying each coefficient separately.

• Linear Equations and Their Applications

  • Linear equations are those with one or more variables of the first order.
  • There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
  • Linear equations can therefore be expressed in general (standard) form as:
  • If the drivers want to designate a meeting point, they can algebraically find the point of intersection of the two functions, as seen in .
  • If the drivers want to designate a meeting point, they can algebraically find the point of intersection of the two functions.

• Introduction to Rational Functions

  • A rational function is any function which can be written as the ratio of two polynomial functions.
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.
  • Singularity occurs when the denominator of a rational function equals $0$, whether or not the linear factor in the denominator cancels out with a linear factor in the numerator.
  • Setting each linear factor equal to zero, we have $x+2 = 0$ and $x-2 = 0$.

• Solving Problems with Logarithmic Graphs

  • Thus, it becomes difficult to graph such functions on the standard axis.
  • Here are some examples of functions graphed on a linear scale, semi-log and logarithmic scales.
  • The top left is a linear scale.
  • That means that if we want to graph a function that is unwieldy on a linear scale we can use a logarithmic scale on each axis and retain the properties of the graph while at the same time making it easier to graph.
  • Similar data plotted on a linear scale is less clear.

• 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.
  • Standard form is another way of arranging a linear equation.
  • In the standard form, a linear equation is written as:
  • 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.
  • Convert linear equations to standard form and explain why it is useful to do so

• Formulas and Problem-Solving

  • Linear equations can be used to solve many everyday and technically specific problems.
  • Linear equations can be used to solve many practical and technical problems.
  • Graph of gratuity as a function of the price of the bill, y=0.18x, where gratuity of 18%.
  • The dependent variable (y) represents gratuity (tip) as a function of cost of the bill (x) before gratuity.
  • Use a given linear formula to solve for a missing variable