Lab 2: Function Plotting (GeoGebra)

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Note: Questions marked (Submit) require you to enter your answer in the Google Form.

Use GeoGebra (https://www.geogebra.org/calculator) to explore graphs, discontinuities, intersections, and transformation sequences, things that are often impossible (or very tedious) to find by algebra alone. If you’re more comfortable with Desmos, feel free to use it, but I think using GeoGebra might be easier to find holes, intersections, and asymptotes.

Problem 1: Functions graphs and properties

In GeoGebra, enter:

$f(x) = 2x^2 - 4x - 16$
$g(x) = x^2 - 5\sqrt{x}$
$h(x) = \frac{x^2+5x+4}{x^3+1}$

Identify for each function (Submit):

Domain
Range
x-axis intercepts

Submit:

a) f-domain, f-range, f-intercepts

b) g-domain, g-range, g-intercepts

c) h-domain, h-range, h-intercepts

Problem 2: Transformations & Recording Graph Features

Start with $f(x) = x^2 - 1$ . In the same GeoGebra view, add:

Transform	Formula	What to Record
a) Vertical shift up 2	$f_1(x) = f(x) + 2$	Find the two $x$ -values where $f_1(x)=5$ . How far apart are they? (i.e. if $f_1(z_1)=f_1(z_2)=5$ , what is $z_2-z_1$ ?)
b) Horizontal shift right 1	$f_2(x) = f(x - 1)$	at $x=2$ , draw the tangent line to $f_2(x)$ . What is the equation of the tangent line?
c) Reflection about the x-axis	$f_3(x) = -\,f(x)$	what is the maximum value that $f_3(x)$ can attain?

Submit: For each $f_i$ , enter the requested feature.

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Problem 3: Multi-Step Word Transformation & Evaluation

Start with $f(x) = \sqrt{x}$ . Apply the following transformations in order to obtain a new function $g(x)$ :

Reflect $f$ across the y-axis.
Shift up by 3 units.
Horizontally stretch by a factor of 2.
Shift left by 1 unit.

Let the final function be $g(x)$ .

Write the final formula: $g(x) = \dots$
Evaluate: What is $g(-4)$ ?

Submit:

a) Final formula $g(x)$

b) Numerical value $g(-4)$

Problem 4: Composition and Domain by Graph

Define:

$u(x) = \sqrt{x}$
$v(x) = \log_{2} x$ (only $x > 0$ )

In GeoGebra, graph $v\bigl(u(x)\bigr) = \log_{2}\bigl(\sqrt{x}\bigr)$ and $u\bigl(v(x)\bigr) = \sqrt{\log_{2} x}$ .
Read off (Submit) domain intervals for each:
- Domain of $\log_{2}\bigl(\sqrt{x}\bigr)$
- Domain of $\sqrt{\log_{2} x}$

Submit:

a) Domain of $\log_{2}\bigl(\sqrt{x}\bigr)$

b) Domain of $\sqrt{\log_{2} x}$

Problem 5: Graph-Only Features in Rational Functions

Enter $r(x) = (x^3 -21 x - 20) / (x + 1)$ $r (x) = (x^{3} - 21 x - 20) / (x + 1)$ .
- Use GeoGebra’s Point tool to mark the hole location.
- What is the y-value of the hole?
Enter $s(x) = (2 x^2 + 2 x - 4) / (x^2 - 3)$ $s (x) = (2 x^{2} + 2 x - 4) / (x^{2} - 3)$ .
- Identify vertical asymptotes (you can use Asymptote(..)).
- Find the horizontal asymptote by graphing a line $y = c$ that the curve approaches.
(If you have the time:) Compare Desmos vs. GeoGebra descriptions: where is each feature easiest to spot?

Submit:

a) Hole in $r(x)$ : location $(x_0, y_0)$

b) Vertical asymptotes of $s(x)$

c) Horizontal asymptote of $s(x)$

Problem 6: Intersection Points and Approximate Solutions

Use GeoGebra to graph both:

$p(x) = 2^{-x}$
$q(x) = x^2 - 1$

Find their intersection points to two decimal places using the Intersect tool.
Why would solving $2^{-x} = x^2 - 1$ algebraically be difficult?

Submit:

Approximate intersection points $(x_1, y_1)$ and $(x_2, y_2)$ up to 2 decimal places.

Additional Resources

GeoGebra Tutorials
Review previous lecture notes on function transformations, discontinuities, and composition.