Question

Write a family equation to model all quadratic functions with y-intercept at (0, -1). Find the specific member of that family that passes through the points (1, -5) and (2, -7)

Answer 1

Family Equation: y = ax² + bx - 1

Specific: y = x² - 5x - 1

Explanation:

Family Equation

Given the y-intercept, the family equation will look like
y = ax² + bx - 1, because the y-intercept is always the Constant
when a polynomial is in standard form.

Specific

Need to find values for a and b that meet the criteria that the graph will pass through (1, -5) and (2, -7).

1. Substitute x and y in the Family equation for Each point:
-5 = a(1)² + b(1) - 1 and -7 = a(2)² + b(2) - 1

2. Simplify and write in Standard Form (for linear equations):
-5 = a + b - 1 and -7 = 4a + 2b - 1
-4 = a + b and -6 = 4a + 2b
-3 = 2a + b

3. Solve the system of equations from step 2 to find a and b:
a + b = -4 ==> b = -a - 4
2a + b = -3 ==> b = -2a - 3
-a - 4 = -2a - 3
a = 1
b = -(1) - 4 = -5

4. Write the Specific Equation with the values of a and b from step 3.