Problem 7 - Precalculus: Functions and Graphs

Question

Sketch the graph of the Inequality. $$x^{2}+1 \leq y$$

Step-by-Step Solution

Answer

Shade the region above and including the parabola y = x^2 + 1 .

Step 1: Understand the problem

We start by interpreting the inequality x^{2}+1 leq y. It means we need to find the set of points (x, y) in the coordinate plane such that y is greater than or equal to x^{2} + 1 .

Step 2: Sketch the boundary curve

The boundary of the inequality is given by the equation y = x^{2} + 1 . This represents a parabola opening upwards with its vertex at the point (0, 1) . Sketch this parabola on the coordinate plane.

Step 3: Identify the region of the inequality

Since the inequality is x^{2}+1 leq y, we want the region above or on the parabola, where y is greater than or equal to x^2 + 1 . Shade this entire region including the parabola itself.

Step 4: Conclusion of the graph

The graph of the inequality shows the shaded region above the parabola y = x^2 + 1 , including the curve itself. This represents all (x, y) points that satisfy the inequality.

Key Concepts

Quadratic Inequality

Quadratic inequalities involve expressions where a quadratic term is compared to another expression, typically using inequality symbols such as \( \leq \), \( \geq \), \( < \), or \( > \). In our example, the inequality is \( x^2 + 1 \leq y \). This means that the values of \( y \) are either greater than or equal to the expression \( x^2 + 1 \). Understanding this concept is crucial for determining which region of the coordinate plane to shade when graphing the inequality.To solve quadratic inequalities:

• Express the inequality in a standard form if needed.
• Identify the boundary condition by setting the inequality to an equation. Here, that's \( y = x^2 + 1 \).
• Determine the direction of shading based on the inequality sign.

The solution is typically a region on the graph rather than just points, illustrating that multiple solutions can satisfy the inequality.

Parabola

The graph of a quadratic equation \( y = ax^2 + bx + c \) is a curve known as a parabola. For our boundary curve \( y = x^2 + 1 \), the parabola opens upward since the coefficient of \( x^2 \) is positive. The shape of the parabola provides crucial information about the region for graphing inequalities.Key features of a parabola:

• Vertex: The highest or lowest point of the parabola. Here, it's at \((0, 1)\).
• Axis of Symmetry: A vertical line passing through the vertex. In this case, it is the line \( x = 0 \).
• Opening Direction: Upward if the leading coefficient is positive, downward if negative.

By sketching the parabola, you help visualize the inequality's solution set and understand the boundary of the shaded region.

Coordinate Plane

A coordinate plane is defined by two perpendicular lines, the x-axis and the y-axis, where we plot points and various graphs. Each point on the plane corresponds to an \( (x, y) \) coordinate pair. For inequalities, the coordinate plane allows us to represent not only equations but also a set of solutions that can be visually analyzed as regions.Here’s how to use the coordinate plane effectively:

• Mark axes with equal intervals to maintain scale.
• Identify the x and y-intercepts if applicable.
• Plot boundary curves or lines such as \( y = x^2 + 1 \) to frame the inequality.

Using the coordinate plane, we can then shade the region representing the set of points that satisfy the inequality, showcasing its solution graphically.

Boundary Curve

In graphing inequalities, the boundary curve represents the threshold between solution and non-solution areas. For the inequality \( x^2 + 1 \leq y \), the boundary is the parabola \( y = x^2 + 1 \). The job of the boundary curve is to:

• Identify the exact limit of the inequality; here, points exactly on \( y = x^2 + 1 \) are considered part of the solution.
• Provide a clear dividing line on the coordinate plane, showing where to begin shading.

This concept helps students understand where they should focus their attention when determining which side of the curve to shade. It's the starting point for representing all potential solutions on the graph.