Problem 20 - Precalculus: Mathematics for Calculus

Question

Graph the inequality. $$y>x^{2}+1$$

Step-by-Step Solution

Answer

Shade above the dashed parabola \( y = x^2 + 1 \), not including the line.

Step 1: Understand the inequality

The given inequality is \( y > x^2 + 1 \). This describes a region in the coordinate plane where the \( y \)-values are greater than the \( y \)-values on the parabola \( y = x^2 + 1 \).

Step 2: Graph the boundary

Before shading the solution area, graph the boundary for the inequality, \( y = x^2 + 1 \). This is a standard parabola that opens upwards with its vertex at the point (0,1). Use a dashed line since the inequality is strict (\( > \), not \( \geq \)).

Step 3: Test a point not on the boundary

Choose a test point that is not on the boundary line to determine the region to shade. A common choice is the origin, (0,0), since it's easy to evaluate. Substitute \((0,0)\) into the inequality: \( 0 > 0^2 + 1 \) which simplifies to \( 0 > 1 \), a false statement. Thus, the region containing the origin is not part of the solution.

Step 4: Shade the correct region

Since the test point in Step 3 was not part of the solution, shade the opposite region. The solution of the inequality \( y > x^2 + 1 \) is the region above the parabola, where \( y \)-values are greater than those on the parabola.

Step 5: Indicate solution region

Make sure to clearly mark the shaded region above the dashed parabola line. This indicates that it includes all points \((x,y)\) such that \( y > x^2 + 1 \).

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of a horizontal axis known as the x-axis and a vertical axis known as the y-axis.
At the intersection of these axes lies the origin, denoted as (0,0). This setup provides a framework for locating and plotting points using ordered pairs \(x, y \).
To place a point on the plane, we move x units along the x-axis and y units along the y-axis.
When graphing equations or inequalities, it's essential to correctly position their corresponding lines or curves on this plane.

• Positive x-values lie to the right of the origin.
• Negative x-values lie to the left.
• Positive y-values rise above the origin.
• Negative y-values fall below.

Understanding this layout allows us to easily visualize where different regions lie, aiding in the effective graphing of inequalities.

Graphing and Analyzing Parabolas

A parabola is a symmetrical, U-shaped curve that's defined by a quadratic equation. In the case of our inequality \( y > x^2 + 1 \), the boundary curve is the standard form of a parabola, \( y = x^2 + 1 \).
This situation specifically calls for graphing the parabola, which has its vertex at (0,1) because the minimum point in this equation is shifted up by 1 unit from the origin.
When sketching \( y = x^2 + 1 \):

• Mark the vertex at (0,1), which is the lowest point of this upwards-opening parabola.
• The parabola is symmetrical around the y-axis.
• Use a dashed line to represent its boundary since the inequality is strict (greater than, not greater than or equal to).

Recognizing these elements helps us distinguish between the parabola itself and the regions defined by inequalities associated with it.

Exploring Inequality Solutions

Inequalities define a range of solutions rather than just one. To solve the inequality \( y > x^2 + 1 \), we first draw the parabola \( y = x^2 + 1 \) using a dashed line, indicating that the points on this line are not included in the solution set.
Next, we choose a test point, often the origin (0,0), to see if it satisfies the inequality, which, in this case, it does not.
This testing allows us to discern the side of the parabola that contains the solutions. If the test point solution is false, as (0,0) was in our exercise, we shade the opposite region.
Analysis of inequality solutions involves:

• Selecting an appropriate test point not on the boundary.
• Checking if this point satisfies the inequality.
• Shading the correct region based on the test result.

Ultimately, graphing the inequality accurately involves recognizing these regions and marking them clearly to visualize where the inequality holds true.