Question

The table of values represents a linear function g(x), where x is the number of days that have passed and g(x) is the balance in the bank account:

x g(x)
0 $325
5 $400
10 $475
Part A: Find and interpret the slope of the function. (3 points)
Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms.
Part C: Write the equation of the line using function notation. (2 points)
Part D: What is the balance in the bank account after 12 days? (2 points)

The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x):
A graph with two linear functions; f of x passes through 1, 3 and 3, 13, and g of x passes through negative 1, 3 and 1, 13.
Part A: Describe two types of transformations that can be used to transform f(x) to g(x).
Part B: Solve for k in each type of transformation. (4 points)
Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x).
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FDK431.10

Answer 1

The slope of the function g(x) is $15 per day, indicating a daily increase in balance. The linear equation can be expressed in various forms using slope, y-intercept, and standard coefficients, with the function notation being g(x) = 15x + 325. After 12 days, the bank account balance is $505.

Step-by-step explanation:

Part A: Finding and Interpreting the Slope

The slope of a linear function represents the rate of change. To find the slope of the function g(x), you calculate the change in y-value divided by the change in x-value between any two points on the line. Using the given table of values, the slope (m) can be calculated using the points (0, $325) and (5, $400):
m = (400 - 325) / (5 - 0) = 75 / 5 = 15. So, the slope is $15 per day, meaning for each additional day, the balance increases by $15.

Part B: Equations of the Line

The point-slope form of the equation is y - y1 = m(x - x1) where m is the slope and (x1, y1) is a point on the line. Using the point (0, $325) and m=15, the equation in point-slope form is y - 325 = 15(x - 0). The slope-intercept form is y = mx + b, which for this line is y = 15x + 325. The standard form of the equation is usually written as Ax + By = C. Converting the slope-intercept form to standard form, you get 15x - y = -325.

Part C: Function Notation

Using function notation, the equation of the line is written as g(x) = 15x + 325.

Part D: Balance after 12 Days

To find the balance after 12 days, substitute x=12 into the function notation equation: g(12) = 15(12) + 325 = 180 + 325 = 505. Therefore, the balance after 12 days is $505.

Transformation of Functions

For the linear functions f(x) and g(x), the transformation from f(x) passing through (1, 3) and (3, 13), to g(x) passing through (-1, 3) and (1, 13), involves a horizontal shift. The values suggest translating f(x) two units to the left to obtain g(x). Alternatively, f(x) could be reflected across the y-axis and then shifted left by 1 unit. Calculating k is not possible without a functional form of f(x). Equations for transformations would be determined based on the type of transformation identified (translation or reflection followed by translation).