Question

Consider the graph given below.


Determine which sequences of transformations could be applied to the parent function, f(x) = x, to obtain the graph above.


Shift left 3 units, reflect over the y-axis, and then vertically stretch by a factor of 2

Shift up 6 units, reflect over the x-axis, and then vertically stretch by a factor of 2 Reflect over the y-axis, vertically stretch by a factor of 2, and then shift up 6 units

Shift left 2 units, reflect over the y-axis, and then vertically stretch by a factor of 6

Shift right 3 units, reflect over the y-axis, and then vertically stretch by a factor of 2

Reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units

Answer 1

Answer:

Reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units.

Step-by-step explanation:

Given:

The parent function is given as:

$f(x)=x$

Now, let us find the equation of the transformed function using the graph.

Slope of the graph is given as the ratio of the y intercept and x intercept.

y-intercept = 6

x-intercept = 3

Therefore, slope is, $m=-\frac{6}{3}=-2$

Negative sign means that the 'y' value decreases as 'x' increases.

Now, equation of a line with slope 'm' and y intercept 'b' is given as:

$g(x)=mx+b$

Plug in -2 for 'm' and 6 for 'b'. This gives,

$g(x)=-2x+6$

So, the parent function $f(x)$ is transformed to $g(x)$.

Now, let us transform the parent function to $g(x)$ using the rules of transformation.

1. Multiplying the parent function by -1.

$f(x)\to -f(x)\\x\to -x$

This is reflection over the x axis.

2. Multiplying $-f(x)$ by 2.

$f(x)\to 2(-f(x))\\-x\to -2x$

This transformation vertically stretches by a factor of 2.

3. Now, add 6 to $2(-f(x))$.

$f(x)\to 2(-f(x))+6\\-2x\to -2x+6$

This shifts the graph up by 6 units.

Therefore, the series of transformations are:

Reflect over the x-axis, vertically stretch by a factor of 2, and then shift up 6 units.