Subjects algebra

Exponential Form 3A1190

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1. **State the problem:** We need to rewrite the given exponential function into the form $$y = ac^x + k$$ where $a$ is the initial value multiplier, $c$ is the base of the exponential, and $k$ is the vertical shift. 2. **Identify the given function:** Since the exact function is not provided, we assume a general exponential function of the form $$y = A \cdot B^x + C$$. 3. **Rewrite the function:** The form $$y = ac^x + k$$ matches $$y = A \cdot B^x + C$$ where $a = A$, $c = B$, and $k = C$. 4. **Explain the parameters:** - $a$ controls the vertical stretch or compression and reflection. - $c$ is the base of the exponential, which must be positive and not equal to 1. - $k$ shifts the graph vertically up or down. 5. **Example:** If the function was $$y = 3 \cdot 2^x + 5$$, it is already in the form $$y = ac^x + k$$ with $a=3$, $c=2$, and $k=5$. Since the exact function is not given, this is the general method to convert any exponential function into the form $$y = ac^x + k$$. **Final answer:** The function rewritten in the form $$y = ac^x + k$$ is $$y = A \cdot B^x + C$$ where $a=A$, $c=B$, and $k=C$.