Question

question:
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.

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 \((x - 1)(x - 3)=0\). 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 \(x = 3\). So the x - intercepts are \((1,0)\) and \((3,0)\).

Step2: Find the vertex (minimum point)

For a quadratic function in the form \(y=ax^{2}+bx + c\) (\(a
eq0\)), the x - coordinate of the vertex 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 two other points

Let's choose two other x - values within the interval \(0\leq x\leq5\).

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

To graph the function:

1. Draw a 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 coefficient of \(x^{2}\) is positive (\(a = 1>0\)), the parabola opens upward. Connect the points with a smooth curve.

Answer:

The key points are \((0,3)\), \((1,0)\), \((2,-1)\), \((3,0)\), \((5,8)\). The graph is a parabola opening upward passing through these points with x - axis labeled "x - value" and y - axis labeled "y - value".