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? Make the answer really short and not advanced.

Answer 1

Note: Question 1 asks for a “quadratic” but the picture and later text show a cubic. I’ll treat it as the cubic shown.

Question 1
a) Equation (standard form)
y = x^3 + 2x^2 − x + 2

b) Table and average rate of change on [−2, 2]

x | f(x)
−2 | 4
−1 | 4
0 | 2
1 | 4
2 | 16

Average rate of change = (f(2) − f(−2)) / (2 − (−2)) = (16 − 4) / 4 = 3

c) Interpretation and use (short)
- The average rate 3 is the slope of the secant line between x = −2 and x = 2.
- The graph is not linear: it increases, then decreases to a local minimum, then increases sharply. Instantaneous rate (slope) changes: zero at the local max and min, negative on the decreasing part, positive elsewhere.
- Real-life use: the average rate tells you the average change over an interval (e.g., average profit change per day over those 4 days), even though daily (instant) changes vary.

Question 2
a) Key features of y = x^3 + 2x^2 − x + 2 (short)
- Domain: all real numbers. Range: all real numbers.
- y-intercept: (0, 2). Two turning points (local max ≈ (−1.55, 4.63) and local min ≈ (0.22, 1.89)). Ends: as x → −∞, y → −∞; as x → ∞, y → ∞.

Comparison to the quadratic from earlier (short)
- Quadratic has one vertex (one turning point) and is symmetric; cubic can have two turning points and changes direction twice.

b) Table and average rate on [−2, 2] for this cubic
(Same table as above)
Average rate = 3.

c) Compare rates and to a linear function (short)
- The average rate 3 is the same secant slope we found earlier for this cubic.
- Unlike a linear function (constant slope everywhere), these nonlinear graphs have rates that change from point to point (instantaneous slope varies). The average rate is just one summary value over an interval.

d) Graph behavior of y = sqrt(x+5) and y = cbrt(x+5) (short)
- Both are shifted left 5 units (start at x = −5).
- y = sqrt(x+5): domain x ≥ −5, starts at (−5,0) and increases slowly (no negative x side).
- y = cbrt(x+5): domain all real numbers, passes through (−5,0) and continues smoothly for x < −5 and x > −5 (S‑shaped, with an inflection at (−5,0)).

e) How root graphs are like/different from quadratic/cubic (very short, simple)
- Similar: all are functions and show increasing or changing behavior; you can talk about slopes and average rates.
- Different: square root is limited to x ≥ −5 (domain restriction) and grows more slowly; cube root has an inflection instead of multiple turning points like a cubic; quadratics/cubics are polynomials with symmetric (quadratic) or multi-turning-point (cubic) shapes and different end behavior.