Algebra 1 · Unit 2
See how changing the slope and y-intercept of y = x tilts, flips, or slides the line.
Jump to section:
The parent is y = x
y = x has slope 1 and y-intercept (0, 0). It is the diagonal through the origin.
Change the slope
y = mx is still through the origin. |m| > 1 is steeper than y = x. 0 < |m| < 1 is flatter. m < 0 flips the line so it falls left to right.
Change the y-intercept
y = x + b slides y = x up (b > 0) or down (b < 0). The slope stays 1.
Both at once
y = mx + b changes the tilt and the starting height. Compare to y = x by asking: steeper or flatter? up or down? flipped?
Compare each graph to y = x.
The slope is still 1.
The y-intercept moves from (0, 0) to (0, 3).
The line slides up 3.
Answer: same slope; up 3
Both go through (0, 0).
Slope 3 is steeper than slope 1.
Answer: steeper; same intercept
Both go through (0, 0).
Slope 12 is flatter than slope 1.
Answer: flatter; same intercept
Both go through (0, 0).
Slope −1 flips the line so it falls left to right.
Answer: flips (falls left to right)
Slope stays 1.
The graph slides down 4. Y-intercept (0, −4).
Answer: down 4
m = 2, b = 5
Steeper than y = x, and up 5.
Answer: steeper and up 5
Describe the change, or write the new equation.
y = mx + b
m = −2, b = 4
y = −2x + 4
Answer: y = −2x + 4
14 is between 0 and 1, so flatter.
b = −3, so down 3.
Answer: flatter and down 3
Up 2 would be y = x + 2.
Slope 5 with that intercept: y = 5x + 2.
Answer: y = 5x + 2
y = −1x has slope −1 and intercept 0, so it is the flip through the origin.
y = 4x is steeper, not flipped. y = x + 4 is a slide.
Answer: y = −1x
Rising means m > 0. Flatter than y = x means 0 < m < 1.
One example: y = 13x
Answer: one answer: y = 13x
Same slope 2, so they are parallel (same steepness).
y = 2x + 3 is the graph of y = 2x slid up 3.
Answer: parallel; second is up 3
Order of changes, and matching a description to y = mx + b.
Steeper first: y = 2x, then down 3: y = 2x − 3.
Slide first: y = x − 3, then slope 2 through that intercept only if you keep b = −3: y = 2x − 3.
If you instead replace x with 2x after the slide, y = 2x − 3 still.
If someone writes 2(x − 3) = 2x − 6, they scaled the intercept too. That is a different change.
The description “change slope to 2 and intercept to −3” is y = 2x − 3. Multiplying the whole slide by 2 is y = 2x − 6.
Answer: y = 2x − 3 if you keep intercept −3; y = 2x − 6 if you scale the slide
Flip: y = −1x
Up 7: y = −1x + 7
Answer: y = −1x + 7
y = 12x + 3
Slope 12 is flatter.
Y-intercept is (0, 3), not a shift of y = x by 6 units up. The +6 was inside, then cut in half.
Answer: y = 12x + 3; up 3, not 6
|−3| > 1, so steeper.
Negative slope, so flipped (falls left to right).
Down 1.
Answer: steeper, flipped, down 1
Parallel to y = 4x means m = 4.
Through (0, −5) means b = −5.
y = 4x − 5
Answer: y = 4x − 5
y = x through (0, 0) and (3, 3).
y = 13x + 2 through (0, 2) and (3, 3).
They meet at (3, 3).
Answer: meet at (3, 3)
y = 2(x − 1) = 2x − 2. Slope 2, y-intercept (0, −2).
y = 2x − 1. Slope 2, y-intercept (0, −1).
“Slope 2 and intercept −1” is y = 2x − 1.
Replacing x by (x − 1) first, then multiplying by 2, is y = 2x − 2, a shift of y = 2x to the right 1 (or down 2).
Answer: 2(x − 1) = 2x − 2; 2x − 1 is different
They meet at (4, 4), so 4 = 4m + b.
Steeper and falling: m < −1 (more negative than −1).
One choice: m = −2. Then 4 = 4(−2) + b, so 4 = −8 + b, b = 12.
y = −2x + 12
Check: −2(4) + 12 = 4.
Answer: one answer: y = −2x + 12
Never meet: same slope as y = x, different intercept. y = x + 5 is parallel to y = x, so they never meet.
Meet once, not at the origin: different slopes. y = 3x − 2 meets y = x when x = 3x − 2, so 2 = 2x, x = 1, point (1, 1).
Answer: parallel: y = x + 5; one meeting: e.g. y = 3x − 2
Need help with transformations of y = x?
Book a Session with Amy