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

consider the quadratic function $y = x^2 - 4x + 3$ for $0 \\leq x \\leq 5$.\n\ngraph the function:\n\nplot 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 $0 \\leq x \\leq 5$.\n\ngraph the function:\n\nplot 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

To find where the graph intersects the x - axis, we set (y = 0). So we solve the equation (x^{2}-4x + 3=0). We can factor the quadratic as ((x - 1)(x - 3)=0). Setting each factor equal to zero gives (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)

For a quadratic function in the form (y=ax^{2}+bx + c), the x - coordinate of the vertex is given by (x=-\frac{b}{2a}). For (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, substitute (x = 2) into the function: (y=(2)^{2}-4\times2 + 3=4-8 + 3=-1). The vertex (minimum point) is ((2,-1)).

Step3: Find two other points

Let's choose (x = 0) and (x = 5) (within the interval (0\leq x\leq5)).

  • When (x = 0): (y=(0)^{2}-4\times0+3 = 3). So the point is ((0,3)).
  • When (x = 5): (y=(5)^{2}-4\times5 + 3=25-20 + 3=8). So the point is ((5,8)).

Step4: Plot the points and draw the parabola

  1. Draw the coordinate plane. Label the x - axis as "x - value" and the y - axis as "y - value".
  2. Plot the points ((1,0)), ((3,0)), ((2,-1)), ((0,3)) and ((5,8)).
  3. Since the parabola opens upward (because (a = 1>0)), draw a smooth curve through these points.

To graph the function:

  • X - intercepts: ((1,0)) and ((3,0))
  • Vertex (minimum): ((2,-1))
  • Other points: ((0,3)) and ((5,8))

Plot these points on a coordinate plane with x - axis labeled "x - value" and y - axis labeled "y - value" and draw a smooth upward - opening parabola through them.