Questions: Given (f(x)=2 x+1), find the value of (k) so that the graph is (y=f(x)+k). [ k= ]

Given (f(x)=2 x+1), find the value of (k) so that the graph is (y=f(x)+k). [ k= ]
Transcript text: Given $f(x)=2 x+1$, find the value of $k$ so that the graph is $y=f(x)+k$. \[ k=\square \]
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Solution

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Solution Steps

Step 1: Identify two points on the line

Two points on the given line are (0, 0) and (2, 4).

Step 2: Find the slope of the line

The slope is calculated as (change in y)/(change in x) = (4-0)/(2-0) = 4/2 = 2.

Step 3: Find the y-intercept

The y-intercept is where the line crosses the y-axis. In this case, the line crosses at the origin (0,0). Therefore, the y-intercept is 0.

Step 4: Write the equation of the line

Using the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we get y = 2x + 0, which simplifies to y = 2x.

Step 5: Compare with f(x) + k

We are given f(x) = 2x + 1. The graph shown is y = 2x. We are looking for a value of k such that y = f(x) + k. Substituting f(x), we get y = 2x + 1 + k. Since the graph is y = 2x, we must have 1 + k = 0.

Step 6: Solve for k

Solving 1 + k = 0 for k, we get k = -1.

Final Answer: The final answer is $\boxed{-1}$

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