sketch the graph of the quadratic function $f(x) = x^2 - 4x + 3$, labeling its vertex, x-intercepts…

Explanation:

Step1: Find the vertex

For a quadratic function (y = ax^{2}+bx + c), the (x)-coordinate of the vertex is (x=-\frac{b}{2a}). Here (a = 1), (b=-4), so (x=\frac{4}{2\times1}=2). Substitute (x = 2) into (y=x^{2}-4x + 3), (y=2^{2}-4\times2 + 3=4-8 + 3=-1). The vertex is ((2,-1)).

Step2: Find the (x)-intercepts

Set (y = 0), then (x^{2}-4x + 3=0). Factor the quadratic: ((x - 1)(x - 3)=0). So (x=1) or (x = 3). The (x)-intercepts are ((1,0)) and ((3,0)).

Step3: Find the (y)-intercept

Set (x = 0), then (y=0^{2}-4\times0+3 = 3). The (y)-intercept is ((0,3)).

Step4: Find the axis of symmetry

The axis of symmetry is (x=-\frac{b}{2a}). Since (a = 1), (b=-4), the axis of symmetry is (x = 2).

To sketch the graph: Plot the vertex ((2,-1)), (x)-intercepts ((1,0)) and ((3,0)), (y)-intercept ((0,3)). Draw a parabola (opens up because (a=1>0)) passing through these points with the axis of symmetry (x = 2).

Answer:

Vertex: ((2,-1)); (x)-intercepts: ((1,0)) and ((3,0)); (y)-intercept: ((0,3)); Axis of symmetry: (x = 2). Sketch a parabola passing through these points.