Question

4 - 21. plot \\( \triangle a b c \\) on graph paper with points \\( a(2,2), b(-2,-2) \\), and \\( c(8,-2) \\).
a. use the function \\( (x, y) \
ightarrow(-1 x,-1 y) \\) to transform \\( \triangle a b c \\). graph and connect the new points then label this triangle \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\). describe how \\( \triangle a b c \\) has been transformed. what, if anything, about the original triangle has been preserved in its image?
b. now use the function \\( (x, y) \
ightarrow(-2 x,-2 y) \\) to transform the original \\( \triangle a b c \\) to create \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\). has \\( \triangle a b c \\) undergone a rigid transformation to create \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\)? what, if anything, about the original triangle has been preserved in its image?
4 - 22. rewrite each of the following products as a sum by using the distributive property.
a. \\( 5 x ( x - 6 ) \\)
b. \\( - 9 y ( 6 - 3 y ) \\)

Explanation:

4 - 22

Part a

Step1: Apply the distributive property \(a(b + c)=ab+ac\)

Here \(a = 5x\), \(b=x\), and \(c=-6\).

$$5x(x - 6)=5x\times x+5x\times(- 6)$$

Step2: Simplify the terms

Using the rule of exponents \(x^m\times x^n=x^{m + n}\) (\(m = 1,n = 1\) for the first - term), we get:

$$5x\times x+5x\times(-6)=5x^{1 + 1}-30x=5x^{2}-30x$$

Part b

Step1: Apply the distributive property \(a(b + c)=ab+ac\)

Here \(a=-9y\), \(b = 6\), and \(c=-3y\)

$$-9y(6-3y)=-9y\times6+(-9y)\times(-3y)$$

Step2: Simplify the terms

For the first - term: \(-9y\times6=-54y\)
For the second - term: Using the rule \(a\times(-b)=-ab\) and \((-a)\times(-b)=ab\) and \(y^m\times y^n=y^{m + n}\) (\(m = 1,n = 1\)), we have \((-9y)\times(-3y)=27y^{1+1}=27y^{2}\)
So, \(-9y(6 - 3y)=27y^{2}-54y\)

Answer:

a. \(5x^{2}-30x\)
b. \(27y^{2}-54y\)