Problem 19 - College Algebra

Question

Graph each inequality. $$ y \geq x^{2}-9 $$

Step-by-Step Solution

Answer

To graph the inequality \(y \geq x^{2}-9\), draw the parabola of \(y = x^{2}-9\) and shade above it to represent the area where the inequality is true.

Step 1: Sketch the parabola

Start by drawing the parabola of the quadratic equation \(y = x^{2}-9\). This is a standard parabola that is shifted 9 units down the y-axis, opening upwards. Place the vertex of the parabola at the point (0,-9) and the y-intercept at the same point. Knowing that the parabola is symmetric, plot a few points on the one side of the parabola and mirror them across the axis of symmetry ( x=0 in this case). These points help to provide a rough sketch of the parabola.

Step 2: Shade the Area

The inequality stated is \(y \geq x^{2}-9\), meaning the y-values are greater than or equal to \(x^{2}-9\). As a result, the shading will be above or on the parabola. Shade the area above the parabola, including the parabola itself, to represent the solution to the inequality.

Key Concepts

Quadratic Functions

Quadratic functions are an essential topic in algebra. These functions are characterized by their highest power being a square, or degree of 2. The standard form of a quadratic function is given by:

• \( y = ax^2 + bx + c \)

Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
The graph of a quadratic function is a curve called a parabola. Parabolas can either open upwards or downwards, depending on the value of \( a \). If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.
Quadratic functions are a foundation for understanding more complex algebraic concepts. They often appear in scenarios involving motion, areas, and optimization problems.
Understanding how to sketch and manipulate parabolas visually gives insight into solving equations and inequalities graphically, like the one in our exercise.

Vertex of a Parabola

The vertex of a parabola is a key point of interest, as it represents the peak or trough of the curve. For the quadratic equation in standard form, \( y = ax^2 + bx + c \), the vertex's x-coordinate is found using the formula:

• \( x = -\frac{b}{2a} \)

Once the x-coordinate is known, substitute it back into the equation to find the y-coordinate, giving the vertex as a point \((x, y)\).
In the exercise example, the equation \( y = x^2 - 9 \) simplifies our work:

• The vertex is at the point \((0, -9)\), directly derived as there is no \( b \) term to consider.

This vertex denotes the lowest point on the parabola, as \( a \) is positive, indicating the parabola opens upwards.
Identifying the vertex helps in sketching the graph accurately and determining the regions relevant for solving inequalities.

Axis of Symmetry

A parabola's axis of symmetry is an imaginary vertical line that passes through its vertex, dividing the parabola into two mirror-image halves. Understanding this feature is crucial for sketching parabolas and solving inequalities.
For the standard quadratic equation \( y = ax^2 + bx + c \), the axis of symmetry can be calculated using the formula:

• \( x = -\frac{b}{2a} \)

In our exercise, the parabola for \( y = x^2 - 9 \) has an axis of symmetry at \( x = 0 \). This is because the \( b \) term is zero, making calculations straightforward.
The axis of symmetry helps in accurately plotting points on one side of the parabola and reflecting them to the other side, ensuring the graph maintains balance and precision.

Shading Regions

Graphing inequalities involves more than plotting the associated function or line; it requires shading a region that satisfies the inequality. For the inequality \( y \geq x^2 - 9 \), the goal is to find all the points \((x, y)\) that satisfy this condition.
To accomplish this, first plot the parabola \( y = x^2 - 9 \). Since the inequality is \( \geq \), it indicates that the solution includes both the parabola itself and the area above it.
Shading the correct region involves understanding the direction indicated by the inequality:

• If \( y \leq \) could have been the sign, you'd shade below the parabola.
• Given \( y \geq \), shade above the parabola, representing all points equal to or greater than the function values.

This technique provides a visual solution to inequalities, making abstract concepts more tangible.