consider the quadratic function $y = x^2 - 4x + 3$ for $0 \\leq x \\leq 5$. graph the function: plot the…

consider the quadratic function $y = x^2 - 4x + 3$ for $0 \\leq x \\leq 5$. graph the function: plot the function $y = x^2 - 4x + 3$ on a coordinate plane. label the x - axis as \x - value\ and the y - axis as \y - value\. mark the coordinates of all key points where the graph intersects the x - axis, reaches its minimum (since the coefficient of $x^2$ is positive, the parabola opens upward), and at least two other points that help show the shape of the parabola within the given interval.

consider the quadratic function $y = x^2 - 4x + 3$ for $0 \\leq x \\leq 5$. graph the function: plot the function $y = x^2 - 4x + 3$ on a coordinate plane. label the x - axis as \x - value\ and the y - axis as \y - value\. mark the coordinates of all key points where the graph intersects the x - axis, reaches its minimum (since the coefficient of $x^2$ is positive, the parabola opens upward), and at least two other points that help show the shape of the parabola within the given interval.

Answer

Explanation:

Step1: Find x - intercepts (set y = 0)

To find where the graph intersects the x - axis, we solve the equation (x^{2}-4x + 3=0). We can factor the quadratic equation as ((x - 1)(x - 3)=0). Setting each factor equal to zero gives us (x-1 = 0) or (x - 3=0), so (x = 1) and (x=3). The x - intercepts are ((1,0)) and ((3,0)).

Step2: Find the vertex (minimum point)

The x - coordinate of the vertex of a quadratic function (y=ax^{2}+bx + c) is given by (x=-\frac{b}{2a}). For the function (y=x^{2}-4x + 3), (a = 1), (b=-4), and (c = 3). So (x=-\frac{-4}{2\times1}=\frac{4}{2}=2). To find the y - coordinate of the vertex, we substitute (x = 2) into the function: (y=(2)^{2}-4\times(2)+3=4 - 8 + 3=-1). So the vertex (minimum point) is ((2,-1)).

Step3: Find additional points

Let's choose some x - values within the interval (0\leq x\leq5) and find the corresponding y - values.

  • When (x = 0): (y=(0)^{2}-4\times(0)+3=3). So the point is ((0,3)).
  • When (x = 4): (y=(4)^{2}-4\times(4)+3=16-16 + 3=3). So the point is ((4,3)).
  • When (x = 5): (y=(5)^{2}-4\times(5)+3=25-20 + 3=8). So the point is ((5,8)).

Graphing Steps:

  1. Draw the coordinate plane: Label the x - axis as "x - value" and the y - axis as "y - value".
  2. Plot the key points:
    • Plot the x - intercepts ((1,0)) and ((3,0)).
    • Plot the vertex ((2,-1)).
    • Plot the additional points ((0,3)), ((4,3)), and ((5,8)).
  3. Draw the parabola: Since the coefficient of (x^{2}) is positive, the parabola opens upward. Connect the points with a smooth curve.

(Note: Since this is a text - based response, we can't actually draw the graph here, but the above steps describe how to plot the function (y=x^{2}-4x + 3) for (0\leq x\leq5).)