For the function \( y=\log _{2}(2 x+4) \), sketch the graph clearly showing the intercepts and asymptote \( [5 \mathrm{mks}] \)

Function by following steps: - step0: Find the \(x\)-intercept/zero: \(y=\log_{2}{\left(2x+4\right)}\) - step1: Set \(y\)=0\(:\) \(0=\log_{2}{\left(2x+4\right)}\) - step2: Swap the sides: \(\log_{2}{\left(2x+4\right)}=0\) - step3: Find the domain: \(\log_{2}{\left(2x+4\right)}=0,x>-2\) - step4: Convert the logarithm into exponential form: \(2x+4=2^{0}\) - step5: Evaluate the power: \(2x+4=1\) - step6: Move the constant to the right side: \(2x=1-4\) - step7: Subtract the numbers: \(2x=-3\) - step8: Divide both sides: \(\frac{2x}{2}=\frac{-3}{2}\) - step9: Divide the numbers: \(x=-\frac{3}{2}\) - step10: Check if the solution is in the defined range: \(x=-\frac{3}{2},x>-2\) - step11: Find the intersection: \(x=-\frac{3}{2}\) Analyze the y intercept of the function \( y=\log _{2}(2 x+4) \) Function by following steps: - step0: Find the y-intercept: \(y=\log_{2}{\left(2x+4\right)}\) - step1: Set \(x\)=0\(:\) \(y=\log_{2}{\left(2\times 0+4\right)}\) - step2: Multiply: \(y=\log_{2}{\left(0+4\right)}\) - step3: Simplify: \(y=2\) Find the vertical asymptotes of \( y=\log _{2}(2 x+4) \). Function by following steps: - step0: Find the vertical asymptotes: \(y=\log_{2}{\left(2x+4\right)}\) - step1: Evaluate the limit: \(\lim _{x\rightarrow -2^{+}}\left(\log_{2}{\left(2x+4\right)}\right)\) - step2: Rewrite the expression: \(\log_{2}{\left(\lim _{x\rightarrow -2^{+}}\left(2x+4\right)\right)}\) - step3: Calculate: \(\log_{2}{\left(0\right)}\) - step4: Calculate: \(-\infty\) - step5: \(x=-2\) is a vertical asymptote\(:\) \(\begin{align}&x=-2\textrm{ }\textrm{is a vertical asymptote}\end{align}\) - step6: List all vertical asymptotes of the function: \(\begin{align}&x=-2\end{align}\) To sketch the graph of the function \( y=\log _{2}(2x+4) \), we need to identify the intercepts and asymptotes. ### Step 1: Find the x-intercept The x-intercept occurs when \( y = 0 \): \[ 0 = \log_{2}(2x + 4) \] This implies: \[ 2x + 4 = 1 \quad \Rightarrow \quad 2x = -3 \quad \Rightarrow \quad x = -\frac{3}{2} \] Thus, the x-intercept is at the point \( \left(-\frac{3}{2}, 0\right) \). ### Step 2: Find the y-intercept The y-intercept occurs when \( x = 0 \): \[ y = \log_{2}(2(0) + 4) = \log_{2}(4) = 2 \] Thus, the y-intercept is at the point \( (0, 2) \). ### Step 3: Find the vertical asymptote The vertical asymptote occurs where the argument of the logarithm is zero: \[ 2x + 4 = 0 \quad \Rightarrow \quad 2x = -4 \quad \Rightarrow \quad x = -2 \] Thus, there is a vertical asymptote at \( x = -2 \). ### Summary of Key Points - **X-intercept**: \( \left(-\frac{3}{2}, 0\right) \) - **Y-intercept**: \( (0, 2) \) - **Vertical asymptote**: \( x = -2 \) ### Sketching the Graph 1. Plot the x-intercept at \( \left(-\frac{3}{2}, 0\right) \). 2. Plot the y-intercept at \( (0, 2) \). 3. Draw a vertical dashed line at \( x = -2 \) to represent the asymptote. 4. The graph approaches the asymptote as \( x \) approaches \(-2\) from the right and increases without bound as \( x \) increases. The graph will look like this: ``` | 3| * 2| (0,2) 1| / | / | / ---|-----/---------------- | / | / | / | / |/ |----------------------- -3 -2 -1 0 1 2 3 ``` This sketch shows the intercepts and the asymptote clearly.