Question

Sketch the graph of the inequality. $$ y \geq x^{2}-3 x+1 $$

Short answer

A sketch of the graph for the inequality involves a parabola with vertex at (1.5, -0.25), open upwards, intersecting the x-axis at points (0.3819, 0) and (2.618, 0) and the shaded region existing above and including the parabola.

Step-by-step solution

Identify the Type and Vertex of the Parabola

The given extended function in a general form \(y = x^2 - 3x + 1\) indicates it's a parabola. The vertex of the parabola could be found using the formula \(-\frac{b}{2a}\), where \(a=1\) and \(b=-3\). This gives us 1.5 as the x-coordinate of the vertex. After substituting \(x = 1.5\) into the original equation, the y-coordinate of the vertex will be -0.25. So the vertex of the parabola is at (1.5, -0.25).

Identify the Axis of Symmetry and x-intercepts

The axis of symmetry of the parabola is the vertical line through the vertex, which is \(x=1.5\). To find the x-intercepts, set \(y=0\) in the original equation, and solve \(x^2-3x+1=0\) for \(x\). This gives the x-intercepts at \(x=0.38\) and \(x=2.62\).

Sketch the Parabola

To start sketching the graph, plot the vertex and x-intercepts. Reflect the points about the axis of symmetry. Since the coefficient of \(x^2\) is positive, the graph opens upwards. Draw a parabola through these points.

Complete the Graph According to the Inequality

Because the inequality is \(y \geq x^{2} - 3x + 1\), the area above the parabola(inclusive) should be shaded. This means the parabola itself should also be included in the shaded region because of the equal sign in the inequality. The final graph includes a shaded region above the parabola graph and includes the parabola itself.