Question

Suppose g(x) = f(x) + k. Identify a value of k that transforms f into g.

Answer 1

Answer:

$k = 5$

Step-by-step explanation:

Given

$g(x) = f(x) + k$

Required

Determine the value of k

First, we need to determine the equation of g(x)

Start by calculating the slope (m)

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Where x and y are two corresponding values:

$(x_1,y_1) = (1,1)$

$(x_2,y_2) = (0,-3)$

The slope (m) is:

$m = \frac{-3- 1}{0 - 1}$

$m = \frac{-4}{- 1}$

$m = 4$

The equation is calculated using:

$y - y_1 = m(x - x_1)$

Where

$m = 4$ and $(x_1,y_1) = (1,1)$

$y - 1 = 4(x - 1)$

$y - 1 = 4x - 4$

Make y the subject of formula

$y = 4x - 4 + 1$

$y = 4x - 3$

So:

$f(x) = 4x - 3$

Next, we determine the equation of f(x)

Start by calculating the slope (m)

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Where x and y are two corresponding values:

$(x_1,y_1) = (-1,-2)$

$(x_2,y_2) = (0,2)$

The slope (m) is:

$m = \frac{2- (-2)}{0 - (-1)}$

$m = \frac{2+2}{0+1}$

$m = \frac{4}{1}$

$m = 4$

The equation is calculated using:

$y - y_1 = m(x - x_1)$

Where

$m = 4$ and $(x_1,y_1) = (-1,-2)$

$y - (-2) = 4(x - (-1))$

$y + 2 = 4(x +1)$

$y + 2 = 4x +4$

Make y the subject of formula

$y = 4x + 4 - 2$

$y = 4x + 2$

So:

$g(x) = 4x + 2$

Recall that

$g(x) = f(x) + k$

Make k the subject

$k = g(x) - f(x)$

Substitute values for g(x) and f(x)

$k = 4x + 2 - (4x - 3)$

$k = 4x + 2 - 4x + 3$

$k = 4x - 4x+ 2 + 3$

$k = 5$

Answer 2

The transformation that maps f into g is a translation transformation,

involving the addition of k to the value of f(x).

• The value of k that transforms f into g is; k = -5

Reasons:

Points on the graph of g(x) are; (0, -3) and (0, 1)

$\mathrm{The \ slope \ of \ the \ graph \ of \ g(x)} =\dfrac{-3 - 1}{0 - 1} = 4$

The equation of g(x) is; y - (-3) = 4·x, which gives;

y = 4·x - 3

The y-intercept = -3

Points on the graph of f(x) are; (-1 -2) and (0, 2)

$\mathrm{The \ slope \ of \ the \ graph \ of \ f(x)} =\dfrac{-2 - 2}{-1 - 0} = 4$

The equation of f(x) is; y - (-2) = 4·(x - (-1)), which gives;

y = 4·x + 4 - 2 = 4·x + 2

The y-intercept = 2

The slopes of the graphs of f(x) and g(x) are equal, therefore, f(x) ║ g(x)

The difference or the transformation that takes the y-intercept of f(x) to the

y-intercept of g(x) is; k = y-intercept(g(x)) - y-intercept(f(x)) = -3 - 2 = -5

Therefore, k = -5

Which gives;

g(x) = f(x) - 5

The value of k that transforms f into g is; k = -5

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