Answer:
The transformations are a reflection across the x-axis and a vertical stretch by a factor of 3. This means that g(x)= −3f(x)
Transformations of Graphs
Many graphs can be formed from a known graph using graph transformations. Some of the most common transformations are below.
• −f(x) is the graph of f(x),but reflected across the y axis.
• f(−x)is the graph of f(x), but reflected across the x axis.
• cf(x) is the graph of f(x), stretched vertically by a factor of c, if c>1or compressed vertically by a factor of c, if 0<c<1.
• cf(x) is the graph of f(x), compressed horizontally by a factor of 1c, if c>1 or stretched horizontally by a factor of 1c, if 0<c<1.
• The graph of f(x)±c is the graph of f(x) shifted vertically up (for +) or down (for -) c units.
• The graph of f(x±c) is the graph of f(x) shifted horizontally left (for +) or right (for -) c units.
Step-by-step explanation:
First, when examining the graphs of f and g notice that the graphs appear to be reflections of each other - but not quite perfectly. The graph of f goes up from left to right while the graph of g goes down from left to right. So there is a reflection across the x-axis.
However, there is also some stretching that is occurring. The graph of both functions contains the point (1,0), but comparing some other points, the graph of f has (0,−1) while the graph of g has the point (0,3). With just a reflection, the graph of g would have (0,1), so it appears to be a vertical stretch by a factor of 3. Checking another point to confirm, the graph of f has a point near (−6,−2). A reflection across the x-axis would result in (−6,2) and a stretch by a factor of 3 would then make the point (−6,6), which matches the graph of g.
So, the transformations are a reflection across the x-axis and a vertical stretch by a factor of 3. This means that g(x)=−3f(x)g(x)=−3f(x)
