Determine the equation of a line from a graph worksheet | Live ... - Free Printable

Let’s solve each problem one by one. For each graph, we need to:

1. Find the slope (m) — how steep the line is. We pick two points on the line and use:
> slope = (change in y) / (change in x) = (y₂ - y₁) / (x₂ - x₁)

2. Find the y-intercept (b) — where the line crosses the y-axis (when x = 0).

3. Write the equation: y = mx + b

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Problem 1

Look at the first graph.

Pick two clear points on the line. Let’s choose:
- Point A: (0, 5) → this is also the y-intercept!
- Point B: (2, 1)

Slope m = (1 - 5) / (2 - 0) = (-4)/2 = -2

Y-intercept b = 5 (since it crosses y-axis at 5)

Equation: y = -2x + 5

✔ Check: When x=1, y should be -2(1)+5 = 3 → looks correct on graph.

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Problem 2

Graph 2.

Pick two points:
- Point A: (0, -5) → y-intercept!
- Point B: (2, -1)

Slope m = (-1 - (-5)) / (2 - 0) = (4)/2 = 2

Y-intercept b = -5

Equation: y = 2x - 5

✔ Check: When x=1, y=2(1)-5=-3 → matches graph.

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Problem 3

Graph 3.

Pick two points:
- Point A: (0, -1) → y-intercept
- Point B: (4, -3)

Slope m = (-3 - (-1)) / (4 - 0) = (-2)/4 = -1/2

Y-intercept b = -1

Equation: y = -½x - 1

✔ Check: When x=2, y = -½(2) -1 = -1 -1 = -2 → looks right on graph.

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Problem 4

Graph 4.

Pick two points:
- Point A: (0, 1) → y-intercept
- Point B: (1, 4)

Slope m = (4 - 1)/(1 - 0) = 3/1 = 3

Y-intercept b = 1

Equation: y = 3x + 1

✔ Check: When x=2, y=3(2)+1=7 → but wait, graph only goes up to y=5? Hmm… let me recheck.

Wait — looking again at Graph 4: The line passes through (0,1) and (1,4)? Actually, from (0,1) to (1,4) is rise 3, run 1 → yes, slope 3. But maybe the grid is scaled differently? No — actually, looking carefully, when x=1, y=4 is too high. Let me pick better points.

Actually, let’s pick:
- (0, 1)
- (2, 7)? That can’t be — graph doesn’t go that high.

Wait — I think I misread. Let me look again.

In Graph 4, the line goes through:
- (0, 1)
- And maybe (1, 4)? But if y-axis only goes to 5, then (1,4) is possible.

But let’s check another point: when x = -1, y = -2? Let’s calculate slope between (0,1) and (-1,-2):

m = (-2 - 1)/(-1 - 0) = (-3)/(-1) = 3 → same.

So yes, slope is 3, y-intercept 1.

Equation: y = 3x + 1

It’s correct — even if the graph cuts off, the math holds.

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Problem 5

Graph 5.

Pick two points:
- Point A: (0, -5) → y-intercept
- Point B: (2, -9)

Wait — let’s find clearer points.

Looking at graph: Line passes through:
- (0, -5)
- Also, when x=1, y=-7? Let’s confirm.

From (0, -5) to (1, -7): change in y = -2, change in x = 1 → slope = -2

Or from (0, -5) to (2, -9): (-9 - (-5))/(2-0) = (-4)/2 = -2 → same.

So slope m = -2

Y-intercept b = -5

Equation: y = -2x - 5

✔ Check: x=1 → y=-2-5=-7 → matches.

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Problem 6

Graph 6.

Pick two points:
- Point A: (0, 1) → y-intercept
- Point B: (4, 3)

Slope m = (3 - 1)/(4 - 0) = 2/4 = 1/2

Y-intercept b = 1

Equation: y = ½x + 1

✔ Check: x=2 → y=1+1=2 → looks correct on graph.

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Final Answer:

1. Slope: -2, y-intercept: 5, Equation: y = -2x + 5
2. Slope: 2, y-intercept: -5, Equation: y = 2x - 5
3. Slope: -1/2, y-intercept: -1, Equation: y = -½x - 1
4. Slope: 3, y-intercept: 1, Equation: y = 3x + 1
5. Slope: -2, y-intercept: -5, Equation: y = -2x - 5
6. Slope: 1/2, y-intercept: 1, Equation: y = ½x + 1