Question

1. the graph of ( f(x)=3^{x} ) is vertically stretched by a factor of ( a ), reflected about the ( x )-axis, moved left 2 and then vertically translated such that the new range is ( (-infty, 4) ), and the ( y )-intercept is -5. determine the equation and sketch the function in the form ( y=aleft(3^{x+h}

ight)+k ). your sketch must include the asymptote and 4 accurate points. (3 marks)

1. the graph below contains the functions: ( f(x)=log x ) and ( y=a log (b(x - h)) ). describe the transformations that occur, in the correct order, when ( y=log x ) is transformed to ( y=a log (b(x - h)) ). (2 marks)

Explanation:

Step1: Analyze vertical stretch and reflection

For the function \(y = 3^{x}\), a vertical stretch by a factor of \(a\) gives \(y=a\cdot3^{x}\), and a reflection about the \(x -\)axis gives \(y=-a\cdot3^{x}\). A horizontal shift left by \(2\) units (using the rule \(y = f(x + h)\) for a shift of \(h\) units to the left) gives \(y=-a\cdot3^{x + 2}\).

Step2: Use \(y -\)intercept to find \(a\)

The \(y -\)intercept occurs when \(x = 0\). We know that \(y=-5\) when \(x = 0\). Substitute \(x = 0\) into \(y=-a\cdot3^{x + 2}+k\).

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The range of \(y = A\cdot b^{x + h}+k\) is \((-\infty,k)\) for \(b>1\) and \(A<0\). Since the range is \((-\infty,4)\), then \(k = 4\).
Substitute \(k = 4\) into \(-5=-9a + k\):

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Step3: Write the final equation

Substitute \(a = 1\) and \(k = 4\) into \(y=-a\cdot3^{x + 2}+k\). The equation of the function is \(y=-3^{x + 2}+4\).

Step4: Find key points

• When \(x=-2\), \(y=-3^{-2 + 2}+4=-1 + 4=3\)
• When \(x=-1\), \(y=-3^{-1+2}+4=-3 + 4 = 1\)
• When \(x = 0\), \(y=-3^{0+2}+4=-9 + 4=-5\)
• When \(x = 1\), \(y=-3^{1+2}+4=-27+4=-23\)

The horizontal asymptote is \(y = 4\) (since for \(y = A\cdot b^{x+h}+k\), the horizontal asymptote is \(y = k\))

1. First, there is a horizontal compression or stretch. If \(b>1\), it is a horizontal compression by a factor of \(\frac{1}{b}\), if \(0 < b<1\), it is a horizontal stretch by a factor of \(\frac{1}{b}\).
2. Then, there is a horizontal translation. If \(h>0\), the graph is shifted to the right by \(h\) units, if \(h < 0\), the graph is shifted to the left by \(|h|\) units.
3. Finally, there is a vertical stretch or compression. If \(|a|>1\), it is a vertical stretch by a factor of \(|a|\), if \(0<|a|<1\), it is a vertical compression by a factor of \(|a|\). If \(a<0\), there is also a reflection about the \(x -\)axis.

Answer:

The equation of the function is \(y=-3^{x + 2}+4\).

For the second - part (question 5):