CurveSketchSteps - Maple Help

Student[Basics]

 CurveSketchSteps
 show steps in the plotting of a specified expression

 Calling Sequence CurveSketchSteps(ex, opts)

Parameters

 ex - expression or string opts - options of the form keyword=value where keyword is one of displaystyle, output

Description

 • The CurveSketchSteps command is used to show how a student might plot a basic expression.
 • This command handles linear and quadratic polynomials in one variable, as well as single trig, log, 1/x and exponentials that are shifted and scaled. It does not support combinations of these functions and works best when the input is already simplified.
 • To avoid automatic simplifications the input expression can be given as a string. For example, calling CurveSketchSteps(cos(x-Pi/2)) will first evaluate cos(x-Pi/2) to sin(x) using normal Maple evaluation rules.  Instead, call CurveSketchSteps("cos(x-Pi/2"), to internally make use of InertForm:-Parse and avoid this simplification.
 • The displaystyle and output options can be used to change the output format.  See OutputStepsRecord for details.
 • This function is part of the Student:-Basics package.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right):$
 > $\mathrm{CurveSketchSteps}\left(2\mathrm{sin}\left(3x+\frac{\mathrm{Pi}}{3}\right)+1\right)$
 $\begin{array}{lll}{}& {}& \text{Let's plot}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2{}\mathrm{sin}{}\left(3{}x+\frac{\mathrm{\pi }}{3}\right)+1\\ \text{•}& {}& \text{Compared to the plot of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\mathrm{sin}{}\left(x\right)\text{, we have a vertical stretch by a factor of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Then, we have a horizontal compression by a factor of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{1}{3}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Then, we have a vertical shift of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}1\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Then, we have a horizontal shift of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}-\frac{\mathrm{\pi }}{9}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Apply the horizontal shift and stretch to the range,}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x=-2{}\mathrm{\pi }..2{}\mathrm{\pi }\\ {}& {}& \left[{}\right]{+}\frac{{\mathrm{\pi }}}{{9}}{..}\left[{}\right]{+}\frac{{\mathrm{\pi }}}{{9}}{=}{-2.443460953}{..}{1.745329252}\\ \text{•}& {}& \text{We can now plot using the information extracted}\\ {}& {}& {\mathrm{PLOT}}{}\left({\mathrm{...}}\right)\end{array}$ (1)
 > $\mathrm{CurveSketchSteps}\left(2{x}^{2}+4x+10\right)$
 $\begin{array}{lll}{}& {}& \text{Let's plot}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2{}{x}^{2}+4{}x+10\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Complete the square}\\ {}& {}& {2}{}{\left({x}{+}{1}\right)}^{{2}}{+}{8}\\ \text{•}& {}& \text{With the expression in vertex form we can extract valuable information}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{The coefficient}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{of the}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\left(x+1\right)}^{2}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{term indicates a parabola that opens}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{up}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{and has a vertical}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{stretch}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{We have a horizontal shift of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}-1\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{and a vertical shift of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}8\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{which gives a vertex of (}-1\text{,}8\text{)}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{We can now plot using the information extracted}\\ {}& {}& {\mathrm{PLOT}}{}\left({\mathrm{...}}\right)\end{array}$ (2)
 > $\mathrm{CurveSketchSteps}\left(4x+10,\mathrm{output}=\mathrm{typeset}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's plot}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}4{}x+10\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{This is a line; find two points and draw a line through them}\\ {}& {}& {y}{=}{4}{}{x}{+}{10}\\ \text{•}& {}& \text{Set}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{= 0 to solve for y intercept}\\ {}& {}& {y}{=}{10}\\ \text{•}& {}& \text{This gives a y intercept of (0,}10\text{)}\\ {}& {}& {y}{=}{10}\\ \text{•}& {}& \text{Set expresson to 0 to solve for}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{intercept}\\ {}& {}& {0}{=}\left[{}\right]\\ \text{•}& {}& \text{Subtract}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}4\cdot x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{from both sides}\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ \text{•}& {}& \text{Simplify}\\ {}& {}& \left[{}\right]{=}{10}\\ \text{•}& {}& \text{Divide both sides by}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}-4\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ \text{•}& {}& \text{Simplify}\\ {}& {}& {x}{=}{-}\frac{{5}}{{2}}\\ \text{•}& {}& \text{This gives an}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{intercept of (}-\frac{5}{2}\text{,0)}\\ {}& {}& {x}{=}{-}\frac{{5}}{{2}}\\ \text{•}& {}& \text{By connecting through the two points we can plot the line}\\ {}& {}& {\mathrm{PLOT}}{}\left({\mathrm{...}}\right)\end{array}$ (3)
 > $\mathrm{CurveSketchSteps}\left(\frac{2}{3x+2},\mathrm{output}='\mathrm{link}'\right)$

Compatibility

 • The Student[Basics][CurveSketchSteps] command was introduced in Maple 2022.