Question

In this unit, you’ve explored different types of functions and their features. Along the way, you may have wondered if you’d ever actually encounter functions or their graphs in the real world. While it’s true that not every career will ask you to graph a square root function, you will certainly benefit from being able to think critically, notice patterns, and implement innovations.

It’s critically important to think of ways to apply new ideas, or innovations, in order to affect the outcome of a given situation. For example, imagine that you own a business and its profits are decreasing. What changes can you make to the functions that model your business? For example, is there a better way to produce your goods and services, or a way to cut costs?

Today, some companies are using the work-from-home model to cut the overhead costs of running a large office space. Due to innovative thinking in response to the COVID-19 pandemic, a strategy that had never been considered in many professional sectors may one day be the norm. Innovations require critical thinking and creativity.

Question 1
Use the image to answer the questions.

A graph of a cubic function.
A two-dimensional graph showing a continuous orange curve. The x-axis ranges from approximately -4 to 4, and the y-axis ranges from -4 to 10. The curve starts low on the left, increases to a local maximum around x = -1.5 and y = 4.5, then decreases to a local minimum around x = 0.5 and y = 2, and then sharply increases as it moves to the right. The curve appears to pass through the point (0, 2).
Write the quadratic equation of the graphed curve in standard form.
Create a table and find the average rate of change over the interval [−2,2]
.
How does the rate of change that you found in Question 1b relate to the graph? Describe the key features of the graph in terms of what happens to the rate of change. Describe a possible use for the average rate of change on this graph. How could you apply it in real life?

































Reveal Answer
Question 2
Use the image to answer the questions.

Graph of a parabola opening upwards on a coordinate plane with vertex at (-0.75, -3.125).
A coordinate plane graph displays a golden-brown parabola that opens upwards. The horizontal x-axis extends from -4 to 4, with tick marks at intervals of 1 and numerical labels at -4, -2, 0, 2, 4. The vertical y-axis extends from -6 to 6, with tick marks at intervals of 1 and numerical labels at -6, -4, -2, 0, 2, 4, 6. The curve intersects the y-axis at approximately positive 0.5. The lowest point of the parabola, its vertex, is clearly marked with a solid black dot and precisely labeled with the coordinates (-0.75, -3.125). Grid lines are present across the entire coordinate plane to assist in reading values.
What are the key features of the cubic function graph y=x3+2x2−x+2
? How does this graph compare to the graph of the quadratic function in the previous questions?
Create a table to find the average rate of change over the interval [–2,2]
.
Compare these rates of change to the rates of change you found for the previous graph (Question 1b). How are they similar or different? How do the graphs of these functions compare to the graph of a linear function?
Graph the equations y=x+5−−−−−√
and y=x+5−−−−−√3
. How are they similar to each other and how are they different?
How are the square root and cube root graphs from Question 2d similar to the quadratic and cubic graphs you’ve been analyzing? What is different about their key features?

Answer 1

Below I answer each question in order. For Question 1 I must make a small clarification: the picture and description show a cubic curve (three turning regions), so I treat Q1 as asking for the cubic equation of the graphed curve (not a quadratic).

Question 1
a. An algebraic cubic matching the graph (using the visible features: local maximum near (−1.5, 4.5), local minimum near (0.5, ~1.54) and the y‑intercept at (0,2)) is
y = (20/27) x^3 + (10/9) x^2 − (5/3) x + 2.
(Equivalently 27y = 20x^3 + 30x^2 − 45x + 54.)

I obtained this cubic by choosing the derivative to have roots at x = −1.5 and x = 0.5 (the observed critical points). That fixes f′(x) up to a constant k; then I used the approximate observed y-value 4.5 at x = −1.5 and the y-intercept 2 at x = 0 to solve for k and the coefficients. (Because the picture gives only approximate values, the coefficients above are an algebraic model that fits those approximate features.)

b. Table and average rate of change over [−2, 2]
Using the cubic above, the function values at integer x are:

x | y = f(x)
−2 | 104/27 ≈ 3.85185
−1 | ? (you can compute if needed)
0 | 2
1 | ?
2 | 244/27 ≈ 9.03704

(For the purposes of the average rate of change on [−2,2] we only need f(−2) and f(2).)

Average rate of change on [−2,2] = [f(2) − f(−2)] / (2 − (−2)) = (244/27 − 104/27) / 4 = (140/27) / 4 = 35/27 ≈ 1.2963.

(If you want the full integer-point table, compute f(−1) and f(1) from the same formula.)

c. How the average rate relates to the graph; key features; real‑life use
- Interpretation: The average rate of change ≈ 1.296 on [−2,2] is the slope of the secant line between x = −2 and x = 2. It measures the net change in y per unit change in x across that whole interval — here, y increases on average by about 1.296 units in y for each 1 unit increase in x across [−2,2].
- How that compares with the graph: the instantaneous rate of change (the slope, f′(x)) is not constant on this cubic. The cubic has
- a local maximum near x ≈ −1.5 (slope = 0 there),
- a local minimum near x ≈ 0.5 (slope = 0 there),
- slopes negative on the interval between the two critical points and positive outside that interval.
So although parts of the curve slope up and down within [−2,2], overall the net effect from x = −2 to x = 2 is an increase (positive average slope).
- Real-life use: the average rate of change tells you the overall trend over an interval even when the instantaneous rate varies. Examples:
- Average growth in revenue per month over a quarter (even if some months dipped).
- Average speed on a trip between two locations (even if speed varied during the trip).
- Average change in production output across a range of inputs.

Question 2
a. Key features of y = x^3 + 2x^2 − x + 2
- Degree and end behavior: cubic (odd degree) → as x → ∞, y → ∞; as x → −∞, y → −∞.
- Critical points: f′(x) = 3x^2 + 4x − 1. Solve 3x^2 + 4x − 1 = 0:
x = [−4 ± sqrt(16 + 12)] / 6 = [−4 ± 2√7] / 6 = (−2 ± √7)/3.
Numerically, the critical x-values are approximately x ≈ −1.5486 and x ≈ 0.2153. These are the locations of the local maximum and local minimum (evaluate f at those x to get the corresponding y-values).
- Inflection: there is one inflection point where f″(x) = 6x + 4 = 0 ⇒ x = −4/6 = −2/3; inflection at x = −2/3.
- y‑intercept: f(0) = 2.

Comparison to the quadratic from Question 2 earlier (vertex at (−0.75, −3.125)):
- A quadratic has exactly one vertex (single turning point) and is symmetric about a vertical axis through the vertex. Its end behavior is the same direction on both ends (both up for positive leading coefficient).
- The cubic has up to two turning points, an inflection point, and opposite end behaviors. So it is qualitatively different (S-shaped) versus the “U-shaped” parabola.

b. Table and average rate of change on [−2,2] for y = x^3 + 2x^2 − x + 2
Compute values:

x | f(x)
−2 | 4 (since −8 + 8 + 2 + 2 = 4)
−1 | 4 (−1 + 2 + 1 + 2 = 4)
0 | 2
1 | 4 (1 + 2 − 1 + 2 = 4)
2 | 16 (8 + 8 − 2 + 2 = 16)

Average rate of change on [−2,2] = [f(2) − f(−2)] / 4 = (16 − 4)/4 = 12/4 = 3.

c. Compare these rates of change to Question 1
- The cubic in Question 2 has average rate of change 3 on [−2,2], which is larger than the average rate ≈ 1.296 found in Question 1’s model. Both are positive, meaning net increasing trend across the interval, but the second cubic rises more rapidly on average.
- Both are cubics and share the “S-shape” behavior (two turning points, one inflection). They differ in the x-locations and y-values of their extrema and in steepness (coefficients), which produces different average rates.

Comparison to a linear function:
- A linear function has constant rate of change (slope) everywhere. For a non-linear function (quadratic, cubic, root functions, etc.), the instantaneous rate of change varies with x. The average rate of change on an interval is just the slope of the secant line between endpoints; it does not describe how the function behaves pointwise inside the interval.

d. Graphs of y = sqrt(x + 5) and y = cbrt(x + 5) — similarities and differences
- Similarities:
- Both are shifts left by 5 units of the basic root graphs.
- Both are increasing functions (their outputs grow as x increases).
- Both have the point (−5, 0) as a reference point (where the inside is zero).
- Differences:
- Domain: y = sqrt(x + 5) has domain x ≥ −5 (square root not real for negative argument), while y = cbrt(x + 5) has domain all real x (cube root of any real is real).
- Shape: sqrt(x + 5) is only the right branch starting at (−5,0), concave down for x > −5 and grows more slowly (slope decreases as x increases). The cube root is oddlike and has an inflection at (−5,0); it extends both left and right and looks like a stretched S (increasing but changes concavity through the inflection).
- End behavior: sqrt grows without bound as x → ∞ but only to the right; cube root grows to ±∞ as x → ±∞ (but slowly).

e. How the square-root and cube-root graphs compare to the quadratic and cubic graphs
- Similarities:
- Square root and quadratic: both have a “turning point” (a minimum) in the sense that the square root’s left end is the lowest point at x = −5 and the parabola has a vertex; both are non-decreasing to the right of that minimum.
- Cube root and cubic: both are oddlike S-shaped in that they pass through an inflection point and change concavity; both are defined for all real x and have opposite end behavior as x → ±∞.
- Differences:
- Degree / symmetry: quadratics and cubics are polynomials with specific symmetry/shape and can have multiple turning points (cubic up to two), while root functions are inverses of even/odd powers and have restricted domains or smoother, slower growth.
- Turning points vs. inflection: the quadratic has a symmetric vertex; the square root has only one endpoint minimum (and no symmetry). The cubic typically has two stationary turning points and an inflection; the cube root has only an inflection (no local maxima/minima), so it never “turns back” — it is monotone increasing everywhere.
- Domains: square-root’s domain is restricted (x ≥ −5), while quadratic’s domain is all reals. Cube root and cubic both have domain all reals, but their rates of change and presence/absence of local extrema differ.

If you want, I can:
- produce a cleaner table of many sample points for any of the functions, or
- plot the four graphs (quadratic, cubic, sqrt, cbrt) together so you can visually compare them, or
- re-fit the cubic in Question 1 to different assumed turning-point y-values if you prefer different approximations from the picture.