General Graphing

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In this activity, you will discover how translations, reflections, and magnifications affect the shape of a graph.

For this activity, you may wish to split yourself into groups to work on the various aspects of each graphing problem.

The Desmos graph below displays the graph of $y=x^2$ , which you will be using in this activity. You can edit this to see the shapes of graphs of other functions. By the end of this activity, you should be able to graph a function that has been translated, reflected, or magnified, knowing what the original function looks like.

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Translations

Graph the following functions by hand or on Desmos to understand how translations affect the shape of a graph.

Group 1: Graph $y = x^2$ , $y = x^2 + 4$ and $y = x^2 - 2$ .
Group 2: Graph $y = x^2$ , $y = (x-3)^2$ and $y = (x+1)^2$ .
Group 3: Graph $y = x$ , $y = x+3$ and $y=x-1$ .

Compare the graphs of $y=(x+5)^2$ and $y=x^2+5$ . Which of the following is a horizontal translation of the graph $y=x^2$ ?

$y=(x+5)^2$ $y=x^2+5$

What affect will adding $D$ to a function value have on a graph? What affect will adding $C$ to the input value ( $x$ ) before applying the function have on the graph?

Interpret $y = x-1$ and $y = (x-1)$ in two different ways and show that their graphs will be the same.

As a group, graph $y = \displaystyle {\frac {1}{x}}$ , $y = \displaystyle {\frac {1}{(x-3)}}$ and $y = \displaystyle {\frac {1}{x} + 2}$ . Which function’s graph is shifted furthest to the right, compared to the graph of $y=\frac {1}{x}$ ?

$y=\frac {1}{x}$ $y=\frac {1}{x-3}$ $y=\frac {1}{x}+2$

If you know what the graph of $y = \sin x$ looks like, can you describe what the graph of $y = (\sin x) + 4$ and $y = \sin (x-\frac {\pi }{4})$ look like?

Reflections

Graph the following functions by hand or with Desmos to understand the effect that relfections have on the shape of a graph.

1.: Group 1: Graph $y = e^x$ , $y = e^{-x}$ , $y=-e^x$ and $y = -e^{-x}$
2.: Group 2: Graph $y = \ln x$ , $y = \ln (-x)$ , $y = - \ln x$ and $y = - \ln (-x)$
3.: Group 3: Graph $y = \sqrt x$ , $y = \sqrt {-x}$ , $y = - \sqrt x$ and $y = - \sqrt {-x}$

Compare the graphs of $y=\sqrt {-x}$ and $y=-\sqrt {x}$ . Which of the following is a vertical reflection of the graph $y=\sqrt {x}$ ?

$y=\sqrt {-x}$ $y=-\sqrt {x}$

What affect will placing a negative sign in front of the function value do to the graph? What affect will placing a negative sign on the input value before applying the function have on the graph?

As a group, graph $y = x^2$ , $y = -x^2$ , $y = (-x)^2$ and $y = -(-x)^2$ . Which function has the same graph as $y=x^2$ ?

$y=-x^2$ $y=(-x)^2$ $y=-(-x)^2$

Why do the graphs of $y = x^2$ and $y = (-x)^2$ look the same? Give two reasons, one by simplifying the second equation algebraically, the second by interpreting the effect of the negative sign on the graph.

If you know what the graph of $y = \sin x$ looks like, can you describe what the graph of $y = - \sin x$ and $y = \sin (-x)$ look like?

Magnifications

Graph the following functions by hand or with Desmos to determine how magnifications affect the shape of a graph.

Group 1: Graph $y = \ln x$ , $y = 4 \ln x$ and $y = \ln {(4x)}$ . (Make special note of where the graph crosses the x-axis.)
Group 2: Graph $y = \sqrt x$ , $y = 2 \sqrt x$ and $y = \sqrt {2x}$ .
Group 3: Graph $y = \frac {1}{x}$ , $y = \frac {2}{x}$ and $y = \frac {1}{2x}$ .

Consider the graph of the function $y=\sqrt {2x}$ . Which of the following functions will give the same graph? Justify your answer algebraically, and by interpreting the affect of magnifications on a graph.

$y=2\sqrt {x}$ $y=\sqrt {2}\sqrt {x}$

What affect will multiplying $A$ to a function value have on a graph? What affect will multiplying $B$ to the input value ( $x$ ) before applying the function have on the graph?

If you know what the graph of $y = \sin x$ looks like, can you describe what the graph of $y = A \sin x$ and $y = \sin (Bx)$ look like?

Summary

Suppose you know the graph of a function $y = f(x)$ and the transformed graph $y = \pm A \cdot f(\pm Bx + C) +D$ .

Which things in the transformation affect the graph horizontally (left and right)?

$\pm A$ $\pm B$ $C$ $D$

Which things in the transformation affect the graph vertically (up and down)?

$\pm A$ $\pm B$ $C$ $D$

How does a multiplier affect the graph? a minus sign? a number added?

If you know the graph of $y = f(x)$ , how can you find the graph of $y = -5 f(4x) - 3$ ? the graph of $y = -5 f(4x + 2) - 3$ ?

Given the graph of $y=f(x)$ , when graphing $y = -5 f(4x + 2) - 3$ above, which transformation should be performed first?

Reflect the graph vertically. Stretch the graph vertically by a factor of $5$ . Compress the graph horizontally by a factor of $4$ . Translate the graph horizontally by $2$ units. Translate the graph vertically by $-4$ units.