graph

Examples of graph in the following topics:

• Visualizing Domain and Range

  • The domain and range can be visualized using a graph, such as the graph for $f(x)=x^{2}$, shown below as a red U-shaped curve.  
  • The range for the graph $f(x)=-\frac{1}{12}x^3$, is $\mathbb{R}$.
  • The graph of $f(x)=x^2$ (red) has the same domain (input values) as the graph of $f(x)=-\frac{1}{12}x^3$ (blue) since all real numbers can be input values.  
  • The range of the blue graph is all real numbers, $\mathbb{R}$.
  • Use the graph of a function to determine its domain and range

• Graphs of Logarithmic Functions

  • Below is the graph of the $y=logx$.
  • The graph crosses the $x$-axis at $1$.
  • That is, the graph has an $x$-intercept of $1$, and as such, the point $(1,0)$ is on the graph.
  • Of course, if we have a graphing calculator, the calculator can graph the function without the need for us to find points on the graph.
  • In fact if $b>0$, the graph of $y=log{_b}x$ and the graph of $y=log{_\frac{1}{b}}x$ are symmetric over the $x$-axis.

• Graphs of Equations as Graphs of Solutions

  • The graph would be a horizontal line through points that all have $y$-values of -4.
  • Similarly, if the equation is $x=C,$ then the graph is a vertical line.
  • The graph of a cubic polynomial has an equation like $y=x^3-9x$.  
  • Therefore, $(-2,10)$ is a point on this curve (i.e., the graph of the equation).
  • Construct the graph of an equation by finding and plotting ordered-pair solutions

• The Vertical Line Test

  • If the vertical line you drew intersects the graph more than once for any value of $x$ then the graph is not the graph of a function.
  • If, alternatively, a vertical line intersects the graph no more than once, no matter where the vertical line is placed, then the graph is the graph of a function.
  • Refer to the three graphs below, $(a)$, $(b)$, and $(c)$.
  • If any vertical line intersects a graph more than once, the relation represented by the graph is not a function.
  • Notice that any vertical line would pass through only one point of the two graphs shown in graphs $(a)$ and $(b)$.

• Graphs of Linear Inequalities

  • Graphing linear inequalities involves graphing the original line, and then shading in the area connected to the inequality.
  • Graphing an inequality is easy.
  • First, graph the inequality as if it were an equation.
  • If the sign is $\leq$ or $\geq$, graph a normal solid line.  
  • Graph of $y\leq x+2$.  

• Basics of Graphing Polynomial Functions

  • A polynomial function in one real variable can be represented by a graph.
  • A typical graph of a polynomial function of degree 3 is the following:
  • This is one thing we can read from the graph.
  • This is again something we can read from the graph.
  • In general, the more function values we compute, the more points of the graph we know, and the more accurate our graph will be.

• Graphs of Exponential Functions, Base e

  • Its graph lies between the graphs of $2^x$ and $3^x$.
  • The graph's $y$-intercept is the point $(0,1)$, and it also contains the point $(1,e).$ Sometimes it is written as $y=\exp (x)$.
  • The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases.
  • The graph always lies above the $x$-axis, but gets arbitrarily close to it for negative $x$; thus, the $x$-axis is a horizontal asymptote.
  • The graph of $e^x$ has the property that the slope of the tangent line to the graph at each point is equal to its $y$-coordinate at that point.

• Tangent as a Function

  • Characteristics of the tangent function can be observed in its graph.
  • The tangent function can be graphed by plotting $\left(x,f(x)\right)$ points.
  • The above points will help us draw our graph, but we need to determine how the graph behaves where it is undefined.
  • At these values, the graph of the tangent has vertical asymptotes.
  • In the graph of the tangent function on the interval $\displaystyle{-\frac{\pi}{2}}$ to $\displaystyle{\frac{\pi}{2}}$, we can see the behavior of the graph over one complete cycle of the function.

• Graphing Equations

  • Equations and their relationships can be visualized in many different types of graphs.
  • Now we can connect the dots to visualize the graph of the equation:
  • This graph is of a parabola (a U-shaped open curve symmetric about a line).
  • This is a graph of a circle with radius 10 and center at the origin.
  • The equation is the graph of a line through the three points found above. 

• Graphical Representations of Functions

  • Extend them in either direction past the points to infinity, and we have our graph.
  • The graph for this function is below.
  • This is the graph of the function $f(x)=5-\frac{5}{2}x$.  
  • Only two points are required to graph a linear function.
  • Graph of the cubic function $f(x)=x^{3}-9x$.