Problem 13 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$y \leq 1-x^{2}$$

Step-by-Step Solution

Answer

Shade the region under the downward-opening parabola \(y = 1 - x^2\), including the parabola itself.

Step 1: Understand the Inequality

To graph the inequality \(y \leq 1 - x^2\), determine the shape of the equation \(y = 1 - x^2\). This equation represents a parabola opening downwards with its vertex at the point \((0, 1)\).

Step 2: Identify the Boundary Line

The boundary of the inequality \(y \leq 1 - x^2\) is the line \(y = 1 - x^2\). Plot this parabola by finding key points: the vertex at \((0, 1)\), and symmetric points like \((-1, 0)\) and \((1, 0)\). Since it opens downwards, for values of \(x\) like \(2\) and \(-2\), calculate the corresponding \(y\)-values to be \(-3\).

Step 3: Determine Shading Direction

Since the inequality is \(y \leq 1 - x^2\), shade the region below the parabola. Points in this region satisfy the inequality, indicating that they are equal to or less than the parabola's path.

Step 4: Draw the Graph

Draw the downward parabola based on the points identified in step 2. Since the inequality includes \(\leq\), draw the parabola as a solid line to indicate that points on the line are included in the solution. Now shade the entire region below this curve to complete the graph of the inequality.

Key Concepts

Parabolas

A parabola is a smooth, U-shaped curve that you can graph on a coordinate plane. It represents a quadratic equation, which is often in the form of \(y = ax^2 + bx + c\). Parabolas can open upwards or downwards, depending on the coefficient of the \(x^2\) term:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

For the inequality \(y \leq 1 - x^2\), you're dealing with a downward opening parabola. This is because the term \(-x^2\) has a negative coefficient. The graph of \(y = 1 - x^2\) has a distinctive U-shape pointing downwards.

Vertex

The vertex is the highest or lowest point on a parabola, acting as a pivotal anchor to the entire curve. It can be found using different methods:

• From the equation \(y = ax^2 + bx + c\), the vertex can be calculated using \(x = -\frac{b}{2a}\).
• Once you have \(x\), plug it back into the equation to solve for \(y\).

In our example of \(y = 1 - x^2\), the vertex can be easily seen as \((0, 1)\) because the equation was already written in the form \(y = a(x - h)^2 + k\), where \(h = 0\) and \(k = 1\). Here, the vertex represents the maximum point on the parabola due to its downward direction.

Shading Regions

Shading regions in graphing inequalities helps visually demonstrate all the solutions to the inequality. For quadratic inequalities, once you have sketched the parabola from the boundary equation, the next step is determining which side of the parabola to shade.

• If the inequality is \(\leq\) or \(<\), like \(y \leq 1 - x^2\), shade below the parabola.
• Conversely, if \(\geq\) or \(>\), shade above.

In this exercise, you shade below the parabola, meaning that all points including and below the curved line of \(y = 1 - x^2\) represent solutions to the inequality \(y \leq 1 - x^2\). This region satisfies the condition that \(y\) must be less than or equal to the value of the parabola at any given \(x\).

Boundary Lines

Boundary lines are critical in graphing inequalities since they physically define the function's limits on the graph. The type of line used can tell whether the boundary itself is included as part of the solution.

• Solid lines are used for \(\leq\) or \(\geq\) inequalities, indicating that points on the line are solutions.
• Dashed lines are for \(<\) or \(>\) inequalities, showing that points on the line are not part of the solutions.

In the inequality \(y \leq 1 - x^2\), you would draw a solid line because \(\leq\) implies that points on the parabola are included in the solution set. It's this construction of boundary lines that visually communicates which points on the plane are possible solutions to the given inequality.