sketch the graph of the quadratic function f(x) = x² - 4x + 3, labeling its vertex, x-intercepts…

Explanation:

Step1: Find the vertex form

To find the vertex form of ( f(x) = x^2 - 4x + 3 ), we complete the square. [ \begin{align*} f(x)&=x^2 - 4x + 3\ &=x^2 - 4x + 4 - 4 + 3\ &=(x - 2)^2 - 1 \end{align*} ] So the vertex form is ( f(x)=(x - 2)^2 - 1 ). The vertex of a parabola in the form ( f(x)=a(x - h)^2 + k ) is ( (h,k) ), so the vertex here is ( (2,-1) ).

Step2: Find the x - intercepts

Set ( f(x)=0 ), so we solve the equation ( x^2 - 4x + 3 = 0 ). Factor the quadratic equation: ( (x - 1)(x - 3)=0 ). Using the zero - product property, if ( ab = 0 ), then either ( a = 0 ) or ( b = 0 ). So ( x - 1 = 0 ) gives ( x = 1 ) and ( x - 3 = 0 ) gives ( x = 3 ). The x - intercepts are ( (1,0) ) and ( (3,0) ).

Step3: Find the y - intercept

Set ( x = 0 ) in the function ( f(x)=x^2 - 4x + 3 ). Then ( f(0)=0^2-4\times0 + 3=3 ). So the y - intercept is ( (0,3) ).

Step4: Find the axis of symmetry

For a quadratic function in the form ( f(x)=ax^2+bx + c ), the equation of the axis of symmetry is ( x=-\frac{b}{2a} ). For ( f(x)=x^2 - 4x + 3 ), ( a = 1 ), ( b=-4 ). So ( x =-\frac{-4}{2\times1}=2 ). The axis of symmetry is the vertical line ( x = 2 ).

To sketch the graph:

• Plot the vertex ( (2,-1) ), the x - intercepts ( (1,0) ), ( (3,0) ), and the y - intercept ( (0,3) ).
• Draw a parabola (since ( a = 1>0 ), it opens upwards) that passes through these points and is symmetric about the line ( x = 2 ).

Answer:

The vertex is ((2, - 1)), the x - intercepts are ((1,0)) and ((3,0)), the y - intercept is ((0,3)), and the axis of symmetry is (x = 2). The graph is a parabola opening upwards with the above - mentioned key points.