Question

given $f(x) = 2x + 4$, describe how the value of $k$ affects the slope and $y$-intercept of the graph of $g$ compared to the graph of $f$.
$g(x) = 2(x + 0.5) + 4$

select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
(type integers or simplified fractions.)
\\(\bigcirc\\) a. the slope is unchanged, and the $y$-intercept is shifted up by \\(\square\\) unit(s).
\\(\bigcirc\\) b. the slope is scaled by a factor of \\(\square\\), and the $y$-intercept is unchanged.
\\(\bigcirc\\) c. the slope and $y$-intercept are both changed. the slope is scaled by a factor of \\(\square\\), and the $y$-intercept is scaled by a factor of \\(\square\\).
\\(\bigcirc\\) d. the slope is unchanged, and the $y$-intercept is shifted down by \\(\square\\) unit(s).
\\(\bigcirc\\) e. the slope and the $y$-intercept are both unchanged.

Explanation:

Step1: Simplify g(x)

First, expand \( g(x) = 2(x + 0.5)+4 \). Using the distributive property \( a(b + c)=ab + ac \), we get \( g(x)=2x+2(0.5)+4=2x + 1+4=2x+5 \).

Step2: Analyze slope and y - intercept of f(x) and g(x)

For \( f(x)=2x + 4 \), the slope \( m_f = 2 \) and the y - intercept \( b_f=4 \) (in the form \( y=mx + b \)).
For \( g(x)=2x+5 \), the slope \( m_g = 2 \) and the y - intercept \( b_g = 5 \).

The slope remains the same (\( m_f=m_g = 2 \)). The y - intercept of \( f(x) \) is 4 and of \( g(x) \) is 5. The change in y - intercept is \( 5 - 4=1 \), so the y - intercept is shifted up by 1 unit.

Answer:

A. The slope is unchanged, and the y - intercept is shifted up by \(\boldsymbol{1}\) unit(s).