7. The graph of \( f(x)=x^{2} \) is transformed as follows: 1. Translated 3 units to the left: This means \( x \) becomes \( (x+3) \). 2. Reflected across the \( y \)-a cis: This means \( x \) becomes \( -x \). 3. Vertically compressed b' a factor of \( \frac{1}{2} \). This means the entire function is multiplied by \( \frac{1}{2} \). 4. Translated 5 units up: Tlis means 5 is added to the entire function. What is the new equation, \( 3(x) \), after these transformations? Answer:

Let's transform the function step by step based on the provided instructions. 1. Start with \( f(x) = x^2 \). 2. Translating 3 units to the left gives us: \( f(x+3) = (x+3)^2 \). 3. Reflecting across the \( y \)-axis results in: \( f(-x+3) = (-x+3)^2 \). 4. Vertically compressing by a factor of \(\frac{1}{2}\): \( \frac{1}{2}((-x+3)^2) \). 5. Finally, translating 5 units up gives us: \( \frac{1}{2}((-x+3)^2) + 5 \). Now, let's simplify: \[ (-x+3)^2 = x^2 - 6x + 9 \] Thus, \[ \frac{1}{2}(x^2 - 6x + 9) + 5 = \frac{1}{2}x^2 - 3x + \frac{9}{2} + 5 \] Note that \(5\) equals \(\frac{10}{2}\), so it can be combined: \[ = \frac{1}{2}x^2 - 3x + \frac{9}{2} + \frac{10}{2} = \frac{1}{2}x^2 - 3x + \frac{19}{2} \] So, the new equation \( g(x) \) after these transformations is: \[ g(x) = \frac{1}{2}x^2 - 3x + \frac{19}{2} \]