Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$y \leq x^{2}-4$$

Step-by-Step Solution

Answer

Shade below \( y = x^2 - 4 \) and include the parabola.

Step 1: Understand the Inequality

The inequality given is \( y \leq x^2 - 4 \). This signifies all the points \((x, y)\) that are below or on the parabola \( y = x^2 - 4 \). The parabola is a basic quadratic equation, with vertex at (0, -4) and opens upwards.

Step 2: Sketch the Parabola

First, we graph the equation \( y = x^2 - 4 \). This is the boundary of the inequality. Since the inequality symbol is \( \leq \), the parabola itself is part of the solution set. Plot points to draw a smooth curve: (0, -4) is the vertex; plot points like (-2, 0), (2, 0), (-1, -3), and (1, -3) to help with sketching the curve.

Step 3: Determine Shading for Inequality

Since we have \( y \leq x^2 - 4 \), shade the area below the parabola. This includes the space beneath the curve and the curve itself because of the \( \leq \) sign, indicating that points on the parabola satisfy the inequality.

Step 4: Draw the Final Graph

On your graph, draw the parabola as a solid line to indicate that points on the line are included in the solution. Then shade the entire region below this parabola. Check sample points like (0, 0) to confirm they satisfy the inequality since 0 is less than -4.

Key Concepts

Quadratic Equations

A quadratic equation is an equation of the form \(y = ax^2 + bx + c\). In our exercise, the equation given is \(y = x^2 - 4\), which is a simple form where \(a = 1\), \(b = 0\), and \(c = -4\). This means our equation is a basic parabolic form without any linear or constant term affecting the slope or position along the y-axis.
Key characteristics of quadratic equations:

• **Parabola Shape:** The graph is a U-shaped curve called a parabola.
• **Vertex:** The point where the curve changes direction, here it's (0, -4).
• **Symmetry:** It is symmetric about the vertical line passing through the vertex.

Understanding quadratic equations helps in assessing how they open, their direction, and their vertex position, contributing to solving inequalities.

Parabolas

Parabolas are special curves that can appear in the graphs of quadratic equations. For \(y = x^2 - 4\), the shape of the equation is a parabola. Here's what you need to know:

• **Standard Form:** A standard parabola looks like \(y = ax^2\). Ours deviates by having a vertical shift down by 4.
• **Direction:** The direction in which a parabola opens is controlled by the coefficient of \(x^2\). A positive coefficient, like 1 in \(x^2\), makes it open upwards.
• **Vertex:** The vertex in this case is at the lowest point (0, -4), meaning it sets the minimum y-value.
• **Axis of Symmetry:** This parabola is symmetric about the y-axis because \(b = 0\) in \(ax^2 + bx + c\).

Understanding how parabolas work can greatly aid in graphing quadratic inequalities like the one in this problem.

Inequality Solutions

Inequality solutions in graphs tell us where one set of coordinates lies in relation to another. In this exercise, the inequality \(y \leq x^2 - 4\) indicates solution coordinates found on or beneath the parabola. Here's how to tackle such inequalities:

• **Boundary Curve:** First, identify the curve that forms the boundary, which is \(y = x^2 - 4\).
• **Inclusion of the Curve:** The \(\leq\) sign shows that this boundary is included. Thus, the graph of the parabola is drawn as a solid line.
• **Shaded Region:** Since the inequality states \(y \leq x^2 - 4\), the shaded region will be under the parabola. All points in this region satisfy the inequality.
• **Verification:** Test points, like (0,0), to ascertain they fit the inequality. For this one, \(0 \leq (0)^2 - 4\) confirms the shading direction.

Mastering inequality solutions through graphing can help visualize which regions of the graph satisfy the given inequality conditions.