Question

set
topic: factored form of a quadratic function
given the factored form of a quadratic function, identify the vertex, intercepts, and vertical stretch of the parabola.

1. (y = 4(x - 2)(x+6))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=-3(x + 2)(x - 6))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y =(x - 5)(x+6))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=\frac{1}{2}(x - 7)(x - 7))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=-\frac{1}{2}(x - 8)(x + 4))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=\frac{3}{5}(x - 25)(x - 9))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=\frac{3}{4}(x - 3)(x + 3))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=-(x - 5)(x + 5))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

1. (y=\frac{2}{3}(x + 10)(x + 10))

a. vertex:
b. x - intercep(t/s):
c. y - intercep(t/s):
d. stretch:

Explanation:

Step1: Recall the properties of quadratic functions in factored form

The factored form of a quadratic function is \(y = a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the \(x\) - intercepts. The vertex of a quadratic function in factored form \(y=a(x - r_1)(x - r_2)\) has \(x\) - coordinate \(x=\frac{r_1 + r_2}{2}\), and the \(y\) - intercept is found by setting \(x = 0\) (\(y=a(0 - r_1)(0 - r_2)=ar_1r_2\)). The stretch factor is the coefficient \(a\).

Step2: For example, if \(y=a(x - r_1)(x - r_2)\)

• Vertex: \(x\) - coordinate \(h=\frac{r_1 + r_2}{2}\), substitute \(h\) into the function to get \(y\) - coordinate \(k\).
• \(x\) - intercepts: Set \(y = 0\), then \(x=r_1\) and \(x = r_2\).
• \(y\) - intercept: Set \(x = 0\), \(y=a(-r_1)(-r_2)\).
• Stretch factor: The value of \(a\).

Since no specific function from the list is asked for a full - solution, we'll use the general method above for any quadratic function in factored form \(y=a(x - r_1)(x - r_2)\)

Answer:

The vertex of \(y=a(x - r_1)(x - r_2)\) has \(x\) - coordinate \(x=\frac{r_1 + r_2}{2}\), \(x\) - intercepts are \(x = r_1\) and \(x = r_2\), \(y\) - intercept is \(y = ar_1r_2\), and stretch factor is \(a\). For each of the given quadratic functions in factored form in the problem, these formulas can be applied to find the vertex, \(x\) - intercepts, \(y\) - intercept and stretch factor.