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When learning horizontal shifts of functions, students are often told that "to shift f() to the right Part A: Determine the formula for a function j() that shifts the function k(x) two units to the left. by c units, you subtract c to get f(x - c)." In contrast, "to shift f(x) to the left c units, you add c to j(x) = get f(x + c)." Have you ever wondered why this non-intuitive "trick" works from a conceptual point of view? Part B: Make sure you can explain your answer in terms of compositions of functions. In other functions that are composed to build j(z)? Part A: We will explore this question by noting that horizontal shifts are actually compositions of Inner function, shifts all z-values two units to the right, m(x)= functions. Let's consider two familiar functions: f(x) - x2 and g(x) = x - 3. What would be the algebraic formula for the composition of the two functions (f o g)(x)? Outer function, k(z) = Question Help: Message instructor (fog)(z) - Calculator Submit Question Part B: We know from Reasoning with Functions 1 that g(x) is the line y - > shifted to the right three units (if we consider the position of the x-intercept), and that f() is a parabola. Given the order of composition, the results should be a parabola that is shifted three units to the right. Construct a table of values for composite function. il in the blanks with the smallest positive number that makes the statement true . . If the graph of sing is shifted unit(s) to the left, it will be the graph of cost. x g(x) = x - 3 f(g(x)) (x, f(g(z))) . If the graph of cost is shifted unit (s)to the right, it will bethe graph of sint. Part B: Which of the following statements is true sin(x) - costx) + O sin (3 - 7 ) - cos(x) D sin(x ) + * - costx ) 3 sin( x + 2) - cos(x ) D cos ( 3 - 7 ) - sinkx 5 Part C: Write the equation for the shifted ve B(x) - Question Help: Message instructor Part C: Plot each of the points in the (x, f(g())) column by hand. If you connect the points, do you Calculator get the parabola described by the algebraic formula derived in Part A? Submit Question O No Yes A researcher has found that a function does a good job of representing her data, except it needs to be shifted 3.5 units to the left along the x-axis. The researcher's function is f(x) = 15.32 + 2.3sin(1.57x). Part A: What is the formula for the adjusted function? Part D: Explain this horizontal shift to a classmate who is still confused. 1 (x) = Edit * Insert " Formats - B U XX AA-