Question

name
perform each of the transformations as described. focus on anchor points!

1. translate the grasshopper right 4 and up 3.
2. reflect the ant over the line y = -x + 2

go
graph each function on the coordinate grid provided.

1. f(x)=2x - 3
2. g(x)=-2x - 3
3. state at least one similarity and one difference between the functions f(x) and g(x).

Explanation:

Step1: Translate the grass - hopper

For a translation right 4 and up 3, if a point \((x,y)\) on the pre - image of the grass - hopper, the new point \((x',y')\) has coordinates \(x'=x + 4\) and \(y'=y+3\). Identify the anchor points of the grass - hopper on the pre - image grid and apply the translation rule to each point.

Step2: Reflect the ant

To reflect a point \((x,y)\) over the line \(y=-x + 2\), first find the equation for the perpendicular line passing through \((x,y)\). The slope of the perpendicular line to \(y=-x + 2\) (slope \(m=-1\)) is \(m' = 1\). The equation of the perpendicular line is \(y - y_0=1\times(x - x_0)\) or \(y=x+(y_0 - x_0)\). Then find the intersection point of \(y=-x + 2\) and \(y=x+(y_0 - x_0)\) by solving the system \(

$$\begin{cases}y=-x + 2\\y=x+(y_0 - x_0)\end{cases}$$

\). Add the two equations: \(2y=2+(y_0 - x_0)\), so \(y = 1+\frac{y_0 - x_0}{2}\) and \(x=1-\frac{y_0 - x_0}{2}\). Let the intersection point be \((a,b)\). Then the reflected point \((x_1,y_1)\) has the property that \((a,b)\) is the mid - point of the line segment connecting \((x,y)\) and \((x_1,y_1)\). Using the mid - point formula \(\frac{x + x_1}{2}=a\) and \(\frac{y + y_1}{2}=b\), we can find \((x_1,y_1)\). Apply this to the anchor points of the ant.

Step3: Graph \(f(x)=2x - 3\)

Find two points on the line. When \(x = 0\), \(y=f(0)=2\times0-3=-3\). When \(y = 0\), \(0=2x-3\), so \(x=\frac{3}{2}\). Plot the points \((0,-3)\) and \((\frac{3}{2},0)\) and draw a straight line through them.

Step4: Graph \(g(x)=-2x - 3\)

When \(x = 0\), \(y=g(0)=-2\times0-3=-3\). When \(y = 0\), \(0=-2x-3\), so \(x=-\frac{3}{2}\). Plot the points \((0,-3)\) and \((-\frac{3}{2},0)\) and draw a straight line through them.

Step5: Compare \(f(x)\) and \(g(x)\)

Similarity: Both \(f(x)=2x - 3\) and \(g(x)=-2x - 3\) have the same \(y\) - intercept, which is \(-3\).
Difference: The slope of \(f(x)\) is \(m_1 = 2\) (it is an increasing function), while the slope of \(g(x)\) is \(m_2=-2\) (it is a decreasing function).

Answer:

For the grass - hopper, new positions of anchor points are obtained by translation. For the ant, new positions of anchor points are obtained by reflection. The graph of \(f(x)\) is a line passing through \((0,-3)\) and \((\frac{3}{2},0)\), the graph of \(g(x)\) is a line passing through \((0,-3)\) and \((-\frac{3}{2},0)\). Similarity: same \(y\) - intercept (\(y=-3\)). Difference: \(f(x)\) has slope \(2\) (increasing), \(g(x)\) has slope \(-2\) (decreasing).