Chapter 8.1, Problem 7QR

Solution

To find: The vertex and axis of the graph of f and describe how the graph f can be obtained from the graph of $g(x)=x^2$ and then graph f .

Vertex : $\left(1,5\right)$

Axis of graph: $x=1$

To obtain $f\left(x\right)$: Shift 1 unit to the right, stretch 3 times along the y -axis, reflect about x -axis and then move 5 units up.

  

Given: $f(x)=3{(x-1)}^2+5$

Concept used:

The vertex form of parabola is given by:-

  $y=a{(x-h)}^2+k$ , where $\left(h,k\right)$ is the vertex.

Th axis of the graph is $x=h.$

If $\left|a\right|>1$ , then the graph is said to be vertically stretched by factor $‘a’$ .

If $0<\left|a\right|<1$ , then the graph is said to be vertically compressed by factor $‘a’$ .

  $-a$ suggest reflection across x -axis.

Calculation:

Comparing the given equation with the standard equation, to get the vertex of the parabola which is $\left(1,5\right)$ and the axis of the graph will be $x=1$ .

To obtain the graph of $f\left(x\right)$ from the graph of $g(x)=x^2$: Shift 1 unit to the right, stretch 3 times along the y -axis and then move 5 units up.