Question

1. use quadratic equation in factored form: f(x)=(x - 3)(x - 1)

a) what is the standard form of given equation?
f(x)=x² - 4x + 3
b) what are the coordinates of the y - intercept? (0, )
c) what are the coordinates of each of the x - intercepts? set each given factor to zero. put answers here: 1) (,0) 2) (,0)
d) what is the x - coordinate of the vertex?
show work here:
x = -b/2a
e) what is the y - coordinate of the vertex?
show work here:
f) sketch the graph on a separate sheet of paper of the given quadratic equation. include the points and coordinates of the vertex, both x - intercepts, y - intercept and the line of symmetry

Explanation:

Step1: Expand to standard form

\[

$$\begin{align*} f(x)&=(x - 3)(x - 1)\\ &=x\times x-x\times1-3\times x + 3\times1\\ &=x^{2}-4x + 3 \end{align*}$$

\]

Step2: Find y - intercept

Set \(x = 0\) in \(f(x)=x^{2}-4x + 3\). Then \(f(0)=0^{2}-4\times0 + 3=3\). The coordinates of the y - intercept are \((0,3)\).

Step3: Find x - intercepts

Set \(x-3 = 0\), then \(x = 3\); set \(x - 1=0\), then \(x = 1\). The coordinates of the x - intercepts are \((1,0)\) and \((3,0)\).

Step4: Find x - coordinate of vertex

For the quadratic function \(y=ax^{2}+bx + c\) (here \(a = 1\), \(b=-4\), \(c = 3\)), the x - coordinate of the vertex is \(x=-\frac{b}{2a}=-\frac{-4}{2\times1}=2\).

Step5: Find y - coordinate of vertex

Substitute \(x = 2\) into \(y=x^{2}-4x + 3\). Then \(y=2^{2}-4\times2+3=4 - 8 + 3=-1\).

Answer:

a) \(f(x)=x^{2}-4x + 3\)
b) \((0,3)\)
c) 1) \((1,0)\) 2) \((3,0)\)
d) \(2\)
e) \(-1\)
f) (Sketching is not possible to show in this text - based format, but you can plot the points \((0,3)\) (y - intercept), \((1,0)\) and \((3,0)\) (x - intercepts), \((2,-1)\) (vertex) and draw a parabola. The line of symmetry is \(x = 2\))