let h(x)=x² - 2x. (a) find the average rate of change from 5 to 7. (b) find an equation of the secant line…

Explanation:

Step1: Calculate ( h(5) ) and ( h(7) )

Given ( h(x)=x^{2}-2x ). For ( x = 5 ), ( h(5)=5^{2}-2\times5=25 - 10=15 ). For ( x = 7 ), ( h(7)=7^{2}-2\times7=49 - 14 = 35 ).

Step2: Calculate the average rate of change

The formula for the average rate of change of a function ( y = h(x) ) from ( x=a ) to ( x = b ) is ( \frac{h(b)-h(a)}{b - a} ). Here ( a = 5 ), ( b = 7 ), ( h(5)=15 ), ( h(7)=35 ). So the average rate of change is ( \frac{h(7)-h(5)}{7 - 5}=\frac{35 - 15}{2}=\frac{20}{2}=10 ).

Step3: Find the equation of the secant line

The slope - intercept form of a line is ( y=mx + c ), where ( m ) is the slope and ( c ) is the ( y ) - intercept. We know the slope ( m = 10 ) (from the average rate of change). Using the point - slope form ( y - y_{1}=m(x - x_{1}) ) with the point ( (x_{1},y_{1})=(5,15) ) (since ( h(5) = 15 )). ( y-15=10(x - 5) ). Expand it: ( y-15=10x-50 ). Add 15 to both sides: ( y = 10x-35 ).

Answer:

(a) The average rate of change from 5 to 7 is ( 10 ). (b) The equation of the secant line is ( y = 10x-35 ).