Algebra 1
Slope-intercept form y = mx + b
This is the form you'll reach for most. Once a line is written as y = mx + b, you can graph it in seconds and read its behavior at a glance, with no table at all. The slope and the starting point are sitting right there in the equation.
Read it as a starting point and a steady pace. The b is where you begin on the y-axis, your starting value, the output before anything happens. The m is your rate of climb from there, the same Unit-3 rate: how much the output changes for each 1 step of input. That's the whole equation: start at b, then climb at rate m.
This reading is clearest in a real situation. Take a phone bill, c = 5t + 30, where t is gigabytes used. The 30 is a $30 base charge you pay before using any data, the starting value, the cost at t = 0. The 5 is $5 per gigabyte, the rate. So you read the equation as "start at $30, add $5 for each gigabyte," not as two loose numbers. In the function form f(x) = mx + b it's the same story: f(0) = b, so feeding 0 into the machine returns the starting value, the intercept.
To graph one by hand, plant the starting point, then walk the slope. Plot (0, b) on the y-axis. Then read the slope as rise over run. For m = 2, that's 2/1, so from the intercept go up 2 and right 1 and mark the next point. Connect them and you have the line. The intercept gives you the first point for free; the slope walks you to the next.
Watch the sign here, because it travels with the number. In y = −3x + 5, the slope is −3, not 3, since the minus is part of it, and the y-intercept is 5. Read the sign that's attached to each number.
New words
- 5.4.d1 Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept, the y-value where the line crosses the y-axis, at the point (0, b).
- 5.4.d2 Function form: f(x) = mx + b, the same line, named as a function. "b is the starting output at x=0; m is how fast the output climbs."
- 5.4.d3 Standard form (recognition): the same line can also be written Ax + By = C (A, B, C integers), for example 2x + y = 5. It hides the slope and intercept, so to read them you solve for y first. Full treatment, plus graphing from intercepts, is Lesson 5.6.
Worked example
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5.4.w1 y = 2x + 1: the slope is m = 2 and the y-intercept is (0, 1). To graph it, plot (0, 1), then go up 2 and right 1 to reach (1, 3), and draw the line.
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5.4.w2 y = -3x + 5: the slope is m = -3 (down 3 for each step right) and the y-intercept is (0, 5). The minus sign is part of the slope, the trap to watch. Since this is the first negative slope you're graphing, work it through: plot the intercept (0, 5), then step down 3, right 1 to (1, 2), and again to (2, −1). Here's the line with those steps shown 5.4.f1.
| x | y = -3x + 5 | point |
|---|---|---|
| 0 | 5 | (0,5) intercept |
| 1 | 2 | (1,2): down 3, right 1 |
| 2 | −1 | (2,-1): down 3 again |
- 5.4.w3 f(x) = (1/2)x - 2: the slope is m = 1/2 and the y-intercept is (0, -2). Reading it as a function: f(0) = -2 (that's the intercept), and f(4) = (1/2)(4) - 2 = 2 - 2 = 0, so (4, 0) is on the line too. A slope of 1/2 means go up 1 for every 2 across.
| x | f(x)=(1/2)x-2 | point |
|---|---|---|
| 0 | −2 | (0,-2) intercept |
| 2 | −1 | (2,-1) |
| 4 | 0 | (4,0) x-intercept |
- 5.4.w4 Read the meaning off the equation. A savings account follows s = 8w + 50, where w is weeks and s is dollars. Identify the slope and intercept and say what each means. The y-intercept is (0, 50): b = 50, so at w = 0 there's already $50 saved, the starting amount. The slope is m = 8: $8 is added each week, the saving rate, $8 per week. So s = 8w + 50 reads "start with $50, add $8 a week." You can evaluate it to predict ahead: s(5) = 8(5) + 50 = 40 + 50 = $90 after 5 weeks. Reading b as the starting value and m as the rate is the move that turns any such equation into a plain-English story.
Here's a clean case to read before the practice turns mixed: in y = 4x − 7, the slope is 4 and the y-intercept is (0, −7). The 4 rides with the x, and the −7 stands alone with its sign. There's nothing more to it.
The most common confusion is mixing up the two roles. The m always rides with the x, and b stands alone, so y = 2x + 1 has slope 2 and intercept 1, never the other way around. A second one: the intercept is a point, (0, b), not just a loose number, which matters the moment you plot it. And if an equation arrives as something like 2x + y = 5, it isn't in this form yet; solve for y first (y = −2x + 5), the same isolating move from Unit 2, before reading the slope and intercept.
Check yourself
- 5.4.c1 In y = -4x + 7, what's the slope and where does the line cross the y-axis? Which way does it tilt? (Slope −4, crossing at (0, 7); the negative slope means it tilts downhill, left to right.)
- 5.4.c2 Write the equation of the line with slope 1/3 and y-intercept (0, -5). (m = 1/3 and b = −5, so y = (1/3)x − 5.)
- 5.4.c3 For f(x) = (1/2)x - 2, what is f(0), and why is that the y-intercept? (f(0) = (1/2)(0) − 2 = −2; it's the y-intercept because the intercept is the output at x = 0, which is exactly f(0).)
- 5.4.c4 A pool drains by d = -20t + 300 (gallons after t minutes). What do the -20 and the 300 mean about the pool? Is it filling or draining, and how fast? (The 300 is the starting amount, 300 gallons at t = 0; the −20 is the rate, losing 20 gallons per minute. The negative rate means it's draining, at 20 gallons a minute.)
You can now read the slope and the y-intercept straight off y = mx + b (and off f(x) = mx + b), graph a line from them by planting the intercept and walking the slope, and read both numbers as a rate and a starting value in a real context.
The practice mixes plain equations with context ones. Answers and the worked examples are right there if one stalls you.
Identify slope and y-intercept:
Reveal answerHide to problem 1
slope 4, y-intercept (0,-7)Reveal answerHide to problem 2
slope -1, y-intercept (0,6)Reveal answerHide to problem 3
slope 2/3, y-intercept (0,1)Reveal answerHide to problem 4
slope 5, y-intercept (0,0)Write the equation given m and b:
Reveal answerHide to problem 5
y = 3x - 4Reveal answerHide to problem 6
y = -2x + 1Find a point on the line (evaluate):
Reveal answerHide to problem 7
2(3)+1 = 7Reveal answerHide to problem 8
-3(2)+5 = -1Reveal answerHide to problem 9
(1/2)(4)-2 = 0Reveal answerHide to problem 10
(1/2)(-2)-2 = -3Interpret in context (state what m and b mean):