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

consider the quadratic function $y = x^2 - 4x + 3$ for $0 \\le x \\le 5$.\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 $0 \\le x \\le 5$.\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 the x-intercepts

Set $y = 0$ and solve $x^2 - 4x + 3 = 0$ by factoring. $$(x - 1)(x - 3) = 0 \implies x = 1, x = 3$$

Step2: Find the vertex (minimum)

Use $x = -\frac{b}{2a}$ for $y = ax^2 + bx + c$. $$x = -\frac{-4}{2(1)} = 2; \quad y = (2)^2 - 4(2) + 3 = -1$$

Step3: Calculate endpoint values

Evaluate the function at the boundaries $x = 0$ and $x = 5$. $$f(0) = 0^2 - 4(0) + 3 = 3; \quad f(5) = 5^2 - 4(5) + 3 = 8$$

Step4: Identify key coordinates

List the points to be plotted on the coordinate plane. $$(1, 0), (3, 0), (2, -1), (0, 3), (5, 8)$$

Answer:

To graph the function $y = x^2 - 4x + 3$ for $0 \leq x \leq 5$:

  1. Draw a coordinate plane with the horizontal axis labeled "x - value" and the vertical axis labeled "y - value".
  2. Plot the x-intercepts at $(1, 0)$ and $(3, 0)$.
  3. Plot the vertex (minimum point) at $(2, -1)$.
  4. Plot the y-intercept/endpoint at $(0, 3)$ and the other endpoint at $(5, 8)$.
  5. Draw a smooth, upward-opening parabolic curve connecting these points within the interval $[0, 5]$.