Question

what feature in geogebra allows for real - time manipulation and observation of graph changes?
a. toolbar
b. input bar
c. slider
d. style bar
which of the following best describes the transformation of the quadratic function ( f(x)=\frac{1}{4}(x - 1)^2+6 )?
a. horizontal shift right by 1, vertical compression by ( \frac{1}{4} ), vertical shift up by 6.
b. horizontal shift right by 1, vertical stretch by 4, vertical shift down by 6.
c. horizontal shift left by 1, vertical stretch by ( \frac{1}{4} ), vertical shift up by 6.
d. horizontal shift left by 1, vertical compression by ( \frac{1}{4} ), vertical shift down by 6.

Explanation:

Step1: Analyze the first question

In GeoGebra, a slider is used for real - time manipulation. For example, if you have a function with a parameter controlled by a slider, moving the slider changes the parameter and the graph changes in real - time.

Step2: Analyze the second question

For the quadratic function \(y = a(x - h)^{2}+k\), the vertex form of a quadratic function. Here \(h = 1\), \(a=\frac{1}{4}\), \(k = 6\).

• The horizontal shift: The formula for horizontal shift is \(y=(x - h)^{2}\). When \(h>0\), the graph shifts to the right. So for \(y=(x - 1)^{2}\), the graph shifts right by 1.
• The vertical transformation: When \(0<|a|<1\) (here \(a = \frac{1}{4}\)), it is a vertical compression.
• The vertical shift: When \(k>0\) (here \(k = 6\)), the graph shifts up by \(k\) units.

Answer:

1. c. Slider
2. a. Horizontal shift right by 1, vertical compression by \(\frac{1}{4}\), vertical shift up by 6.