Q. Which graph represents \(y – 1 = 2(x – 2)\)?

Answer

Convert to slope-intercept form:
\[
y-1=2(x-2)\implies y-1=2x-4\implies y=2x-3.
\]
So the graph is a straight line with slope \(2\) and y-intercept \((0,-3)\). It passes through \((2,1)\) (and through \((1,-1)\), \((0,-3)\), etc.).

Final: the line \(y=2x-3\) (slope \(2\), y-intercept \(-3\)).

Detailed Explanation

Solution (step-by-step)

Write the given equation:
\[ y – 1 = 2(x – 2) \]
Expand the right-hand side:
\[ y – 1 = 2x – 4 \]
Isolate \(y\) to get slope-intercept form:

Add 1 to both sides to obtain \( y = 2x – 3 \).
Identify slope and y-intercept:

The equation \( y = 2x – 3 \) has slope \( m = 2 \) and y-intercept \( b = -3 \) (point \( (0, -3) \)).
Find another point to plot:

Using the slope (rise/run = \( 2/1 \)) from \( (0, -3) \), move right 1 and up 2 to get point \( (1, -1) \). The given point-slope form also shows the line passes through \( (2, 1) \).
Locate the x-intercept (optional check):

Set \( y = 0 \): \( 0 = 2x – 3 \), so \( x = \frac{3}{2} = 1.5 \). The x-intercept is \( (1.5, 0) \).
Conclusion – which graph represents the line:

The correct graph is the straight line that crosses the y-axis at \( (0, -3) \) and rises with slope \( 2 \), passing through points such as \( (0, -3) \), \( (1, -1) \), and \( (2, 1) \). Any graph matching those features represents the equation \( y – 1 = 2(x – 2) \).

See full solution

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FAQs

Q What form is the equation in and what does it tell you?

A The equation \(y-1=2(x-2)\) is point-slope form; it gives slope \(m=2\) and a point \((2,1)\) the line passes through.

Q How do I convert it to slope-intercept form?

A Expand and solve for \(y\): \(y-1=2x-4\) so \(y=2x-3\). Slope-intercept form is \(y=2x-3\).

Q What are the x- and y-intercepts?

A y-intercept: from \(y=2x-3\) is \((0,-3)\). x-intercept: set \(y=0\): \(0=2x-3\) gives \(x=3/2\), so \((3/2,0)\).

Q How do I sketch the graph quickly?

A Plot the point \((2,1)\), then use slope 2: from that point go right 1, up 2 to get another point. Connect with a straight line and extend both ways.

Q How do I check if a given graph matches this line?

A Verify it passes through \((2,1)\), has y-intercept \((0,-3)\), and slope of 2 (rise 2/run 1). If all match, it's the correct graph.

Q What does slope 2 mean in words?

A It means for every 1 unit you move right along the x-axis, the y-value increases by 2 units. The line rises steeply to the right.

Q What is the standard (general) form of the line?

A Rearranging \(y=2x-3\) gives \(2x-y-3=0\), or equivalently \(2x-y=3\).

Q Which lines are parallel or perpendicular to this one?

A Parallel lines have slope 2. Perpendicular lines have slope \(-\tfrac{1}{2}\).

Q What are the domain and range of this line?

A Both domain and range are all real numbers, since a nonvertical line continues indefinitely in both directions.

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