Recognizing the graph of a linear equation from slope and intercept

Key takeaway: This item checks whether you can identify the graph of a line from its slope and y-intercept.

Use slope and intercept

The equation is $y = 3x - 2$. In slope-intercept form, $y = mx + b$, the slope is $m = 3$ and the y-intercept is $b = -2$.

That means the correct graph must cross the y-axis at $(0,-2)$ and rise 3 units for every 1 unit moved to the right. A line with positive slope this steep will tilt upward clearly from left to right.

What the graph should look like

Start at the y-intercept $-2$. Then use the slope $3 = \frac{3}{1}$:

• move right 1
• move up 3

So another point on the line is $(1,1)$. If you continue, you get $(2,4)$ and so on. The graph is a straight line increasing quickly.

This means the correct choice is the graph that crosses the y-axis at $-2$ and rises steeply to the right.

How to eliminate wrong choices

Any graph with a negative slope is impossible, because the coefficient of $x$ is positive. Any graph that crosses the y-axis at a value other than $-2$ is also wrong. If a graph is too flat, its slope is not 3. If it rises but not steeply enough, it does not match the equation.

Key properties to remember

The slope tells direction and steepness. The y-intercept gives the starting point on the vertical axis. For $y = 3x - 2$, the line goes up as x increases and starts two units below the origin.

When you see an equation in slope-intercept form, the fastest method is to read off $m$ and $b$, then match those values to the graph options.

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Pitfalls the pros know 👇 A common mistake is to focus only on the negative intercept and ignore the slope. Another mistake is to treat the coefficient 3 as if it were the y-intercept. In slope-intercept form, the number multiplying x is the slope, and the constant term is the intercept. Also, some students choose a graph that crosses the y-axis at -2 but slopes downward, which cannot represent a positive slope equation.

What if the problem changes? If the equation were $y = -3x - 2$, the graph would still cross the y-axis at -2, but it would slope downward instead of upward. If it were $y = 3x + 2$, the line would rise the same way but cross the y-axis at 2. If the equation were written as $2y = 6x - 4$, dividing by 2 would give the same line: $y = 3x - 2$.

Tags: slope-intercept form, y-intercept, linear graph