Question

Write the equations for f(x) and g(x). Then identify the reflection that transforms the graph of f(x) to the graph of g(x).

Answer 1

From the graph, let's write the equations for f(x) and g(x).

Apply the slope-intercept form of linear equations:

y = mx + b

Where m is the slope and b is the y-intercept.

• For f(x):

To find the slope apply the slope formula:

`m=(y2-y1)/(x2-x1)`

Take two points on the line of f(x):

(x1, y1) ==> (0, -1)

(x2, y2) ==> (-2, 0)

Thus, we have:

`\begin{gathered} m=(0-(-1))/(-2-0) \\ \\ m=(0+1)/(-2) \\ \\ m=-(1)/(2) \end{gathered}`

The y-intercept (b), is the point the line crosses the y-axis.

Therefore, the y-intercept for f(x) is at b = -1

Therefore, the equation for f(x) is:

`f(x)=-(1)/(2)x-1`

• For g(x):

Take two points on the line of g(x).

(x1, y1) ==> (-2, 0)

(x2, y2) ==> (2, 2)

Apply the slope formula to find the slope:

`\begin{gathered} m=(y2-y1)/(x2-x1) \\ \\ m=(2-0)/(2-(-2)) \\ \\ m=(2)/(2+2) \\ \\ m=(2)/(4) \\ \\ m=(1)/(2) \end{gathered}`

The y-intercept of g(x) is: b = 1

Therefore, the equation for g(x) is:

`g(x)=(1)/(2)x+1`

To determine the reflection that transforms f(x) to g(x):

From the graph, we can see the red line represents f(x) while the blue line is that of g(x).

Apply the transformation rules for functions.

When f(x) becomes -f(x), there is a reflection over the x-axis.

Here, the parent function f(x) becomes -f(x) = g(x)

`-(1)/(2)x-1=-(-(1)/(2)x-1)=(1)/(2)x+1`

Therefore, there was a reflection over the x-axis that transforms the grah f(x) to g(x).

ANSWER:

• f(x) = -½x - 1

,

• g(x) = ½x + 1

• Reflection over the x-axis.