(Solved): 2. Equations of lines, function notation, graphing, difference quotients. (a) Find the equation of ...

2. Equations of lines, function notation, graphing, difference quotients. (a) Find the equation of the line passing through the points \( (-2, f(-2)) \) and \( (3, f(3)) \) if \( f(x)=x^{2} \). Sketch a graph illustrating the function \( f(x)=x^{2} \) and the secant line you found. (b) Find the equation of the line passing through the points \( (-2, f(-2)) \) and \( (-2+ \) \( h, f(-2+h)) \) if \( f(x)=x^{2} \). What does the \( h \) represent? Confirm your solution to part (a) by selecting an appropriate value for \( h \). (c) Sketch a graph illustrating the function \( f(x)=x^{2} \) and two secant lines passing through the point \( (a, f(a)) \) if \( a=-2 \). For one secant line, let \( h>0 \), for the other secant line, let \( h<0 \). (d) Find the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x)=x^{2} \) and simplify it. What does the \( x \) represent? What does the \( h \) represent? What does the \( (x+h) \) represent? Confirm your solution to part (a) by selecting appropriate values for \( x \) and \( h \). (e) If we let \( x=-2 \) and \( h=-1 \), the simplified difference quotient from part (d) gives the slope for which line? Find the equation of this new secant line and graph it, along with the graph of \( f(x)=x^{2} \).