Summary of Sketching an equation, Symmetry Test and Exercises

• Sketching an equation: you usually will be given either '' y being expressed in term of x" or ''x being expressed in term of y"

Type I: When y is being expressed in terms of x, for example Then we treat x as inputs and y as outputs when we make the x-y table:

Can you sketch the graph now? [plot(sqrt(x-4),x=0..7, y=0..10); ]

Type II. When x being expressed in term of y. Then we treat y as inputs and x as outputs when we make the x-y table. For example, x=2y²-1, then

Can you sketch the graph now? Notice that you should get a graph that is symmetric to the x-axis. Use

• Exercises: Sketch the following graphs by making the x-y table. Use Maple to check your answers.

1. y=3x-2 [maple syntax: plot(3*x-2, x=-5..5, y=-10..10); )
2. y=x²-2 [maple syntax: plot(x^2-2, x=-5..5);]
3. y=-2x²+1 [maple syntax: plot(-2*x^2+1, x=-5..5);]
4. [maple syntax: plot(abs(x), x=-5..5);]
5. 
6. [maple syntax: plot(sqrt(x+1), x = -2..3);]
7. [maple syntax: plot(sqrt(x)-2, x= -5..5); ]

Note that if we would like to plot the graphs of #5, #7 and #8 together, the maple syntax is:

plot({sqrt(x), sqrt(x+1), sqrt(x)-2}, x=-5..5);

• Symmetry Tests: The purpose of knowing these tests is that they allow us to see how the graph should roughly look like before we actually sketch graphs.

1. Type I. Symmetric to the x-axis: If both (x,y) and (x,-y) both satisfy an equation, then the graph of such equation is symmetric to the x-axis. For example, x=y².
2. Type II. Symmetric to the y-axis: If both (x,y) and (-x,y) both satisfy an equation, then the graph of such equation is symmetric to the y-axis. For example, y=x².
3. Type III. Symmetric to the origin, (0,0): If both (x,y) and (-x,-y) both satisfy an equation, then the graph of such equation is symmetric to the origin, (0,0). For example, y=x³.

• Exercises: Determine if the following equations are symmetric to the x-axis, y-axis or the origin. (Use Maple to check your answers)

1. [ plot(2*abs(x), x=-5..5); ]
2. [ plot(abs(x) + 1, x =-3..3, y=0..3); ]
3. [ plot(abs(x+1), x= -3..1); ]
4. [ plot(1/x , x= -10..10, y=-5..5); ]
5. y=4x²
6. 
7. x²-y²=1 [ with(plots): implicitplot(x^2 - y^2 =1, x =-2..2, y=-2..2, scaling=constrained); ] (note that you need to also copy "with(plots):"
8. x²+y²=1 [ with(plots): implicitplot(x^2 + y^2 =1, x =-2..2, y=-2..2, scaling=constrained); ] (note that you need to also copy "with(plots):"