Question

if $g(x)=f(x)+k$, then what is the value of $k$ ?
equations for 2 functions are shown.
$f(x)=-4x - 10$ $g(x)=-4x + 2$
a vertical shift up of how many units will map $f(x)$ onto $g(x)$?

Explanation:

Step1: Analyze the function transformation

We know that \( g(x)=f(x)+k \), and we have \( f(x)= - 4x-10 \) and \( g(x)=-4x + 2 \). Substitute \( f(x) \) into the equation \( g(x)=f(x)+k \), we get \( -4x + 2=(-4x-10)+k \).

Step2: Solve for \( k \)

Simplify the equation: \( -4x + 2=-4x-10 + k \). Cancel out \( - 4x \) on both sides, we have \( 2=-10 + k \). Then, add 10 to both sides: \( k=2 + 10=12 \)? Wait, no, wait the first part of the problem (the graph part) or the second part? Wait, the second part: \( f(x)=-4x - 10 \), \( g(x)=-4x+2 \). To find the vertical shift, we can also look at the y - intercepts. The y - intercept of \( f(x) \) is \( - 10 \) (when \( x = 0 \), \( f(0)=-10 \)), and the y - intercept of \( g(x) \) is \( 2 \) (when \( x = 0 \), \( g(0)=2 \)). The vertical shift \( k \) is \( g(0)-f(0)=2-(-10)=12 \)? Wait, but maybe I made a mistake. Wait, the first problem (graph) and the second (equations). Wait, the user's problem has two parts? Wait, the first part: "If \( g(x)=f(x)+k \), then what is the value of \( k \)?" with a graph, and the second part: "Equations for 2 functions are shown. \( f(x)=-4x - 10 \), \( g(x)=-4x + 2 \). A vertical shift up of how many units will map \( f(x) \) onto \( g(x) \)?"

Wait, let's solve the second part. For a vertical shift, if \( g(x)=f(x)+k \), then \( k \) is the vertical shift. So \( g(x)=f(x)+k\Rightarrow -4x + 2=(-4x - 10)+k\Rightarrow k=2+10 = 12 \). Wait, but maybe the first part (graph) was miswritten. Wait, maybe the first part's graph: let's assume \( f(x) \) and \( g(x) \) are linear functions. Let's take the y - intercepts. Suppose \( f(x) \) has y - intercept \( b_1 \) and \( g(x) \) has y - intercept \( b_2 \). Then \( k=b_2 - b_1 \). If in the graph, \( f(x) \) crosses y - axis at, say, \( - 4 \) and \( g(x) \) at \( 4 \), then \( k = 8 \), but the second part with equations: \( f(x)=-4x-10 \), \( g(x)=-4x + 2 \). So the vertical shift is \( 2-(-10)=12 \). Wait, maybe the user's first part (the 8) was a wrong answer, and the second part is to find the vertical shift.

Wait, let's focus on the second part: \( f(x)=-4x - 10 \), \( g(x)=-4x + 2 \). We can write \( g(x)=f(x)+k \), so \( -4x + 2=-4x-10 + k \). Subtract \( -4x \) from both sides: \( 2=-10 + k \), so \( k = 12 \). So the vertical shift up is 12 units.

Answer:

12