which is the graph of f(x) = x² - 2x + 3?

Explanation:

Step1: Identify the form of the quadratic function

The function $f(x)=x^{2}-2x + 3$ is in the form $y = ax^{2}+bx + c$ where $a = 1$, $b=-2$, $c = 3$. Since $a=1>0$, the parabola opens upwards.

Step2: Find the x - coordinate of the vertex

The formula for the x - coordinate of the vertex of a quadratic function $y=ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. Substituting $a = 1$ and $b=-2$ into the formula, we get $x=-\frac{-2}{2\times1}=1$.

Step3: Find the y - coordinate of the vertex

Substitute $x = 1$ into the function $f(x)=x^{2}-2x + 3$. Then $f(1)=1^{2}-2\times1 + 3=1 - 2+3=2$. So the vertex of the parabola is $(1,2)$.

Step4: Find the y - intercept

Set $x = 0$ in the function $f(x)=x^{2}-2x + 3$. Then $f(0)=0^{2}-2\times0 + 3=3$. So the y - intercept is 3.

The graph with a parabola opening upwards, vertex at $(1,2)$ and y - intercept at 3 is the correct one. Without seeing all the options, we can say that the graph which has these characteristics is the answer. If we assume the top - shown graph has these features:

Answer:

The top - shown graph (assuming it has a parabola opening upwards, vertex at $(1,2)$ and y - intercept at 3)