Asked by Lemy
UNIT 1 Patterns within Functions
● Given the parent functions for linear, exponential, and quadratic
functions, create a graph and state the domain and range for each.
● Represent any linear function with a table, graph, equation, and visual
model.
● Represent any exponential function with a table, graph, equation, and
visual model.
● Represent any quadratic function with a table, graph, equation, and visual model.
● How do the rates of change for linear, exponential, and quadratic functions compare to each other? Parent Functions
Linear 𝑦 = 𝑥
Exponential 𝑦 = 2௫ Quadratic 𝑦 = 𝑥
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GPT-5 mini
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Below is a compact guide for the parent functions y = x (linear), y = 2^x (exponential), and y = x^2 (quadratic), plus one example of each type shown by table, equation, graph description, and a simple visual model. At the end I compare their rates of change.
1) Parent functions — graphs, domain, range
- Linear: y = x
- Graph: straight line through the origin with slope 1 (45° line).
- Domain: (−∞, ∞)
- Range: (−∞, ∞)
- Key points: (−2, −2), (−1, −1), (0, 0), (1, 1), (2, 2)
- Exponential: y = 2^x
- Graph: increasing curve, passes through (0, 1), horizontal asymptote y = 0 on the left.
- Domain: (−∞, ∞)
- Range: (0, ∞)
- Key points: (−3, 1/8), (−2, 1/4), (−1, 1/2), (0, 1), (1, 2), (2, 4)
- Quadratic: y = x^2
- Graph: parabola opening upward with vertex at (0, 0), symmetric about the y-axis.
- Domain: (−∞, ∞)
- Range: [0, ∞)
- Key points: (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)
2) Representations for one example of each type
A. Linear example: y = 2x + 1
- Table (x → y):
- x = −2 → y = −3
- x = −1 → y = −1
- x = 0 → y = 1
- x = 1 → y = 3
- x = 2 → y = 5
- Graph description: straight line, y-intercept (0,1), slope = 2 (rise 2 for every run 1).
- Visual model: number-line steps or counters: each increase of x by 1 adds 2 more counters to y (show stacked columns increasing by equal increments). Also drawing a staircase connecting points shows constant rise.
B. Exponential example: y = 3·2^x
- Table (x → y):
- x = −2 → y = 3/4 = 0.75
- x = −1 → y = 3/2 = 1.5
- x = 0 → y = 3
- x = 1 → y = 6
- x = 2 → y = 12
- Graph description: increasing exponential curve, passes through (0,3), approaches y = 0 as x → −∞, grows rapidly for positive x.
- Visual model: doubling blocks scaled by 3: start with 3 blocks at x=0, then double each step: 3, 6, 12, 24… (a picture of stacks doubling each step shows multiplicative growth).
C. Quadratic example: y = (x − 1)^2 + 2
- Table (x → y):
- x = −1 → y = 6
- x = 0 → y = 3
- x = 1 → y = 2 (vertex)
- x = 2 → y = 3
- x = 3 → y = 6
- Graph description: parabola opening up with vertex at (1, 2), symmetric about the vertical line x = 1.
- Visual model: square arrays: for x values measured from the vertex, y − 2 equals the square of that horizontal distance. Another visual: growth of areas (1×1, 2×2, 3×3) or showing that successive increases follow odd-number additions (1, 3, 5, 7…).
3) How their rates of change compare
- Linear functions:
- Rate of change is constant (the slope). Example y = 2x + 1 has a constant change of +2 in y for each +1 in x.
- First differences (for equal x-steps) are constant.
- Quadratic functions:
- Rate of change is not constant; it changes at a steady rate itself.
- First differences are not constant, but second differences (differences of first differences) are constant. Example y = x^2 at x = 0,1,2,3,4 gives y = 0,1,4,9,16. First differences: 1,3,5,7; second differences: 2,2,2 (constant).
- Exponential functions:
- Change is multiplicative (constant ratio), not additive. For base 2, each increase of x by 1 multiplies y by 2 (constant growth factor).
- Successive ratios y(x+1)/y(x) are constant (for y = a·b^x, ratio = b).
- First differences grow (or shrink) at an accelerating rate; second differences are not constant.
- Comparison of long-term growth:
- For large x, exponential growth (b^x with b>1) outpaces any polynomial (including quadratics) and linear growth. Example at x = 0..6:
- y = x (linear): 0, 1, 2, 3, 4, 5, 6
- y = x^2 (quadratic): 0, 1, 4, 9, 16, 25, 36
- y = 2^x (exponential): 1, 2, 4, 8, 16, 32, 64
- At small x a quadratic can be larger than an exponential, but eventually the exponential dominates.
Quick checklist for graphing each parent function
- Linear y = mx + b: plot y-intercept (0,b), then use slope m (rise/run) to draw a straight line. Domain and range are all real numbers.
- Exponential y = a·b^x (b>0): plot (0,a), remember horizontal asymptote at y=0 (if no vertical shift), domain all real numbers, range depends on a and vertical shifts (positive if a>0).
- Quadratic y = ax^2 + bx + c: find vertex, axis of symmetry, y-intercept c, plot symmetric points. Domain all real numbers, range is [vertex y-value, ∞) if a>0 (or (−∞, vertex y-value] if a<0).
If you want, I can:
- Draw these graphs as images,
- Make printable visual models (tiles or block diagrams) for classroom use, or
- Generate worksheets with tables and blank graphs for practice. Which would you like next?
1) Parent functions — graphs, domain, range
- Linear: y = x
- Graph: straight line through the origin with slope 1 (45° line).
- Domain: (−∞, ∞)
- Range: (−∞, ∞)
- Key points: (−2, −2), (−1, −1), (0, 0), (1, 1), (2, 2)
- Exponential: y = 2^x
- Graph: increasing curve, passes through (0, 1), horizontal asymptote y = 0 on the left.
- Domain: (−∞, ∞)
- Range: (0, ∞)
- Key points: (−3, 1/8), (−2, 1/4), (−1, 1/2), (0, 1), (1, 2), (2, 4)
- Quadratic: y = x^2
- Graph: parabola opening upward with vertex at (0, 0), symmetric about the y-axis.
- Domain: (−∞, ∞)
- Range: [0, ∞)
- Key points: (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)
2) Representations for one example of each type
A. Linear example: y = 2x + 1
- Table (x → y):
- x = −2 → y = −3
- x = −1 → y = −1
- x = 0 → y = 1
- x = 1 → y = 3
- x = 2 → y = 5
- Graph description: straight line, y-intercept (0,1), slope = 2 (rise 2 for every run 1).
- Visual model: number-line steps or counters: each increase of x by 1 adds 2 more counters to y (show stacked columns increasing by equal increments). Also drawing a staircase connecting points shows constant rise.
B. Exponential example: y = 3·2^x
- Table (x → y):
- x = −2 → y = 3/4 = 0.75
- x = −1 → y = 3/2 = 1.5
- x = 0 → y = 3
- x = 1 → y = 6
- x = 2 → y = 12
- Graph description: increasing exponential curve, passes through (0,3), approaches y = 0 as x → −∞, grows rapidly for positive x.
- Visual model: doubling blocks scaled by 3: start with 3 blocks at x=0, then double each step: 3, 6, 12, 24… (a picture of stacks doubling each step shows multiplicative growth).
C. Quadratic example: y = (x − 1)^2 + 2
- Table (x → y):
- x = −1 → y = 6
- x = 0 → y = 3
- x = 1 → y = 2 (vertex)
- x = 2 → y = 3
- x = 3 → y = 6
- Graph description: parabola opening up with vertex at (1, 2), symmetric about the vertical line x = 1.
- Visual model: square arrays: for x values measured from the vertex, y − 2 equals the square of that horizontal distance. Another visual: growth of areas (1×1, 2×2, 3×3) or showing that successive increases follow odd-number additions (1, 3, 5, 7…).
3) How their rates of change compare
- Linear functions:
- Rate of change is constant (the slope). Example y = 2x + 1 has a constant change of +2 in y for each +1 in x.
- First differences (for equal x-steps) are constant.
- Quadratic functions:
- Rate of change is not constant; it changes at a steady rate itself.
- First differences are not constant, but second differences (differences of first differences) are constant. Example y = x^2 at x = 0,1,2,3,4 gives y = 0,1,4,9,16. First differences: 1,3,5,7; second differences: 2,2,2 (constant).
- Exponential functions:
- Change is multiplicative (constant ratio), not additive. For base 2, each increase of x by 1 multiplies y by 2 (constant growth factor).
- Successive ratios y(x+1)/y(x) are constant (for y = a·b^x, ratio = b).
- First differences grow (or shrink) at an accelerating rate; second differences are not constant.
- Comparison of long-term growth:
- For large x, exponential growth (b^x with b>1) outpaces any polynomial (including quadratics) and linear growth. Example at x = 0..6:
- y = x (linear): 0, 1, 2, 3, 4, 5, 6
- y = x^2 (quadratic): 0, 1, 4, 9, 16, 25, 36
- y = 2^x (exponential): 1, 2, 4, 8, 16, 32, 64
- At small x a quadratic can be larger than an exponential, but eventually the exponential dominates.
Quick checklist for graphing each parent function
- Linear y = mx + b: plot y-intercept (0,b), then use slope m (rise/run) to draw a straight line. Domain and range are all real numbers.
- Exponential y = a·b^x (b>0): plot (0,a), remember horizontal asymptote at y=0 (if no vertical shift), domain all real numbers, range depends on a and vertical shifts (positive if a>0).
- Quadratic y = ax^2 + bx + c: find vertex, axis of symmetry, y-intercept c, plot symmetric points. Domain all real numbers, range is [vertex y-value, ∞) if a>0 (or (−∞, vertex y-value] if a<0).
If you want, I can:
- Draw these graphs as images,
- Make printable visual models (tiles or block diagrams) for classroom use, or
- Generate worksheets with tables and blank graphs for practice. Which would you like next?
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