Chapter 1: Algebra Review
Chapter 1 Using Technology
1.3 Using Desmos to graph a linear function or system
Graphing a Linear Function
Graph[latex]\,f\left(x\right)=-\frac{2}{3}x+5.[/latex]
Show Solution
To graph the line in Desmos:
- Type the equation into the equation 1 box, pictured in blue, always using y= instead of f(x)=.
- Use the [latex]\fbox{ / }[/latex] to type a fraction and right arrow [latex]\fbox{ → }[/latex] to exit the denominator before typing the x.
Graphing a System of Equations
Solving a System of Equations in Two Variables by Graphing
Solve the following system of equations by graphing. Identify the type of system.
Show Solution
Simply type the first equation into equation 1 and the second into equation 2. Desmos will show the point of intersection as a gray point that can be clicked on to see the ordered pair.
The solution to the system is the ordered pair[latex]\,\left(-3,-2\right),[/latex].
Using Desmos to graph functions
The graph of the parent function is changed by factors such as:
- vertical and horizontal shifts
- reflections about the x-axis and y-axis
- compressions and stretches
- and combinations of the above
You can get a better idea of how these factors change the appearance of the graph with these tools:
Graphing Combined Vertical and Horizontal Shifts of Absolute Value
Given [latex]\,f\left(x\right)=|x|,\,[/latex] sketch a graph of [latex]\,h\left(x\right)=|x+1|-3.[/latex]
To graph a function in Desmos:
- Expand the keyboard at the bottom of the screen.
- Type the equation into the equation 1 box, pictured in blue.
- type y= instead of function notation
- use the [latex]\boxed{|a|}[/latex] button at the bottom to open and close the absolute value signs
- You can delete the first equation (X) and type a new one. Note the [latex]\boxed{a^2}[/latex] sign for graphing a parabola and the [latex]\boxed{\sqrt{ }}[/latex] sign for graphing a square root.
Using Desmos to graph Quadratic, Polynomial, and Rational Equations
1.5 Graphing Quadratic functions
Quadratic functions can be graphed with this Desmos Graphing parabolas tool.
1.6 Graphing Polynomial Equations
Graphing a Polynomial Function
We can use technology to find the intercepts of[latex]\,f\left(x\right)={x}^{3}-5{x}^{2}-x+5[/latex] by graphing instead of by factoring.
Show Solution
Type the equation into Desmos. Then click on each of the x-intercepts to see it's coordinates.
1.6 Graphing Rational Functions
Graphing a Rational Function
Sketch a graph of[latex]\,f\left(x\right)=\frac{\left(x+2\right)\left(x-3\right)}{{\left(x+1\right)}^{2}\left(x-2\right)}.[/latex]
Show Solution
Type the equation into Desmos. Click on a critical point to see it's coordinates. The graph makes it easy to visualize the vertical asymptotes.
1.7 Using Desmos to graph Exponential and Logarithmic Equations
Sketching the Graph of an Exponential Function of the Form f(x) = bx
Sketch a graph of[latex]\,f\left(x\right)={0.25}^{x}.\,[/latex]
Show Solution
Type the equation into Desmos. y = 0.25, [latex]\boxed {a^{b}}[/latex], (x)
1.7 Graphing Logarithmic Functions
Graphing a Logarithmic Function of the Form f(x) = logb(x)
Graph[latex]\,f\left(x\right)={\mathrm{log}}_{5}\left(x\right).\,[/latex]
Show Solution
Type the equation into Desmos. y = [latex]\boxed {functions}[/latex], CALCULUS, [latex]\boxed {log_a}[/latex], (x)
