compare the graphs of g(x) to their parent function f(x). state the transformations that have been performed…

compare the graphs of g(x) to their parent function f(x). state the transformations that have been performed on g(x) from the parent function f(x). 1. g(x)=e^{x - 7}+6; f(x)=e^{x} 2. g(x)=-\frac{1}{2}e^{x + 10}; f(x)=e^{x}

compare the graphs of g(x) to their parent function f(x). state the transformations that have been performed on g(x) from the parent function f(x). 1. g(x)=e^{x - 7}+6; f(x)=e^{x} 2. g(x)=-\frac{1}{2}e^{x + 10}; f(x)=e^{x}

Answer

Explanation:

Step1: Analyze horizontal shift for $g(x)=e^{x - 7}+6$

For a function $y = f(x - h)+k$, the horizontal shift is given by $h$. Here $h = 7$, so there is a right - shift of 7 units.

Step2: Analyze vertical shift for $g(x)=e^{x - 7}+6$

The vertical shift is given by $k$. Here $k = 6$, so there is an upward shift of 6 units.

Step3: Analyze $g(x)=-\frac{1}{2}e^{x + 10}$

For $y = af(x + h)$, the factor $a=-\frac{1}{2}$ causes a vertical stretch by a factor of $\frac{1}{2}$ and a reflection across the $x$ - axis. The value $h=- 10$ causes a left - shift of 10 units.

  1. For $g(x)=e^{x - 7}+6$ compared to $f(x)=e^{x}$:
    • Horizontal shift: 7 units to the right.
    • Vertical shift: 6 units up.
  2. For $g(x)=-\frac{1}{2}e^{x + 10}$ compared to $f(x)=e^{x}$:
    • Horizontal shift: 10 units to the left.
    • Vertical stretch by a factor of $\frac{1}{2}$ and reflection across the $x$ - axis.

Answer:

  1. Horizontal shift 7 units right, vertical shift 6 units up.
  2. Horizontal shift 10 units left, vertical stretch by factor $\frac{1}{2}$ and reflection across $x$ - axis.