Question

sketch the graph of the quadratic function $f(x) = x^2 - 4x + 3$, labeling its vertex, x-intercepts, y-intercept, and axis of symmetry.

Explanation:

Step1: Find vertex x-coordinate

For $f(x)=ax^2+bx+c$, vertex $x=-\frac{b}{2a}$.
Here $a=1, b=-4$, so $x=-\frac{-4}{2\times1}=2$.

Step2: Find vertex y-coordinate

Substitute $x=2$ into $f(x)$.
$f(2)=2^2-4\times2+3=4-8+3=-1$

Step3: Find x-intercepts

Set $f(x)=0$, solve $x^2-4x+3=0$.
Factor: $(x-1)(x-3)=0$, so $x=1$ and $x=3$.

Step4: Find y-intercept

Set $x=0$, solve $f(0)$.
$f(0)=0^2-4\times0+3=3$

Step5: Find axis of symmetry

Axis is $x=-\frac{b}{2a}$, same as vertex x.
$x=2$

Answer:

• Vertex: $(2, -1)$
• x-intercepts: $(1, 0)$ and $(3, 0)$
• y-intercept: $(0, 3)$
• Axis of symmetry: $x=2$

(To sketch the graph: Plot the intercepts and vertex, draw a upward-opening parabola symmetric about $x=2$, passing through all labeled points.)