1. Transformations. For the function \( f(x)=x^{2}+2 x \), apply the given transformations in the given order. After each transformation, write the equation describing the new function. Also, give the final location (after applying all transformations) of the reference points \( (-1,-1),(0,0) \), and \( (1,3) \). Shift 3 units to the right: NEW FUNCTION Shift 2 units down: Reflect about the \( y \)-axis: NEW FUNCTION Final location of reference points: WORK: Indicate the algebraic replacements used to find each new function, and show the intermediate points.

Let's start with the function \( f(x) = x^2 + 2x \). ### Step 1: Shift 3 units to the right To shift a function \( f(x) \) to the right by \( h \) units, we replace \( x \) with \( x - h \). Here, \( h = 3 \). \[ f(x) \rightarrow f(x - 3) = (x - 3)^2 + 2(x - 3) \] Now, let's simplify this: \[ f(x - 3) = (x - 3)^2 + 2(x - 3) = (x^2 - 6x + 9) + (2x - 6) = x^2 - 4x + 3 \] **NEW FUNCTION after shifting 3 units to the right:** \[ f_1(x) = x^2 - 4x + 3 \] ### Step 2: Shift 2 units down To shift a function down by \( k \) units, we subtract \( k \) from the function. Here, \( k = 2 \). \[ f_1(x) \rightarrow f_1(x) - 2 = (x^2 - 4x + 3) - 2 = x^2 - 4x + 1 \] **NEW FUNCTION after shifting 2 units down:** \[ f_2(x) = x^2 - 4x + 1 \] ### Step 3: Reflect about the \( y \)-axis To reflect a function about the \( y \)-axis, we replace \( x \) with \( -x \). \[ f_2(x) \rightarrow f_2(-x) = (-x)^2 - 4(-x) + 1 = x^2 + 4x + 1 \] **NEW FUNCTION after reflecting about the \( y \)-axis:** \[ f_3(x) = x^2 + 4x + 1 \] ### Final location of reference points Now, we will find the final locations of the reference points \( (-1, -1) \), \( (0, 0) \), and \( (1, 3) \) after applying all transformations. 1. **For the point \( (-1, -1) \)**: - Shift 3 units to the right: \[ (-1 + 3, -1) = (2, -1) \] - Shift 2 units down: \[ (2, -1 - 2) = (2, -3) \] - Reflect about the \( y \)-axis: \[ (-2, -3) \] 2. **For the point \( (0, 0) \)**: - Shift 3 units to the right: \[ (0 + 3, 0) = (3, 0) \] - Shift 2 units down: \[ (3, 0 - 2) = (3, -2) \] - Reflect about the \( y \)-axis: \[ (-3, -2) \] 3. **For the point \( (1, 3) \)**: - Shift 3 units to the right: \[ (1 + 3, 3) = (4, 3) \] - Shift 2 units down: \[ (4, 3 - 2) = (4, 1) \] - Reflect about the \( y \)-axis: \[ (-4, 1) \] ### Final Locations of Reference Points: - The final location of \( (-1, -1) \) is \( (-2, -3) \). - The final location of \( (0, 0) \) is \( (-3, -2) \). - The final location of \( (1, 3) \) is \( (-4, 1) \).