Question

explore & reason
the table represents a(x), the area of a square as a function of side length x units, where x is a positive real number.
side length (units) | x | 1 | 2 | 3 | 4
model | □ | □ | □ | □ |
area (sq units) | a(x) | 1 | 4 | 9 |

a. consider the function where the areas in the table are doubled. write the equation of a function that represents this.

b. look for relationships graph the ordered pairs for both a(x) and your new function. how would you describe the differences in the locations of these points?

c. find the equation for a function whose x-values are the same as a(x) but whose y-values are 2 units greater than each y-value in a(x).

habits of mind
communicate precisely do you think the effect of altering a(x) in the two ways you did would work for any function you start with? explain.

Explanation:

Step1: Identify base function A(x)

The area of a square with side length $x$ is $A(x) = x^2$.

Step2: Solve Part A: Double the area

Multiply the base function by 2:
$g(x) = 2A(x) = 2x^2$

Step3: Solve Part B: Describe graph differences

For each $x$, the new function's $y$-value is 2 times $A(x)$'s $y$-value. So each point $(x, 2x^2)$ is a vertical stretch of $(x, x^2)$ by a factor of 2.

Step4: Solve Part C: Add 2 to y-values

Add 2 to the base function:
$h(x) = A(x) + 2 = x^2 + 2$

Step5: Solve Habits of Mind

Vertical stretches (multiplying output) and vertical shifts (adding to output) are general transformations that apply to all functions, not just quadratic ones. For any function $f(x)$, $kf(x)$ stretches it vertically by factor $k$, and $f(x)+c$ shifts it up by $c$ units.

Answer:

A. $g(x) = 2x^2$
B. Each point of the new function is a vertical stretch of the corresponding point on $A(x)$ by a factor of 2 (the y-coordinate is twice as large for the same x-coordinate).
C. $h(x) = x^2 + 2$
Habits of Mind: Yes, these effects work for any function. Multiplying a function's output by a constant creates a vertical stretch/compression, and adding a constant to the output creates a vertical shift; these are universal function transformations, not limited to quadratic functions like $A(x)$.