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? (0,3)
c) what are the coordinates of each of the x - intercepts? set each given factor to zero. put answers here: 1) (1,0) 2) (3,0)
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: Recall quadratic standard form

The factored - form of a quadratic is $f(x)=(x - 3)(x - 1)$. Expand it using FOIL method: $(x-3)(x - 1)=x\times x+x\times(-1)+(-3)\times x+(-3)\times(-1)=x^{2}-x - 3x + 3=x^{2}-4x + 3$.

Step2: Find y - intercept

The y - intercept is found by setting $x = 0$ in the function $y=f(x)$. When $x = 0$, $y=f(0)=0^{2}-4\times0 + 3=3$. So the y - intercept is at the point $(0,3)$.

Step3: Find x - intercepts

The x - intercepts are found by setting $y = f(x)=0$. Given $f(x)=(x - 3)(x - 1)=0$. Using the zero - product property, if $ab = 0$, then $a = 0$ or $b = 0$. So $x-3=0$ gives $x = 3$ and $x - 1=0$ gives $x = 1$. The x - intercepts are $(1,0)$ and $(3,0)$.

Step4: Find x - coordinate of the vertex

For a quadratic function in standard form $y=ax^{2}+bx + c$ (here $a = 1$, $b=-4$, $c = 3$), the x - coordinate of the vertex is given by the formula $x=-\frac{b}{2a}$. Substitute $a = 1$ and $b=-4$ into the formula: $x=-\frac{-4}{2\times1}=\frac{4}{2}=2$.

Step5: Find y - coordinate of the vertex

Substitute $x = 2$ into the function $y=x^{2}-4x + 3$. Then $y=2^{2}-4\times2+3=4-8 + 3=-1$.

Answer:

a) $f(x)=x^{2}-4x + 3$
b) $(0,3)$
c) $(1,0)$ and $(3,0)$
d) $2$
e) $-1$
f) (Sketching is not possible to show here, but you would 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 $x = 2$.)