Parabolic Inequalities: Graphing y = a(x – p)^2 + q

Welcome back to Math 11! This week, we’re taking our mathematical journey to the next level by diving into the realm of parabolic inequalities. We’ll be exploring how to graph equations of the form y = a(x – p)^2 + q, where ‘a’, ‘p’, and ‘q’ dictate the shape, position, and orientation of the parabola. Get ready to delve into greater than, less than, greater than or equal to, and less than or equal to as we unravel the mysteries of graphing parabolic inequalities.

Understanding Parabolic Inequalities

Before we leap into graphing, let’s refresh our memory on parabolic inequalities. In the equation y = a(x – p)^2 + q.

‘a’ determines the direction and width of the parabola, ‘p’ shifts the graph horizontally, and ‘q’ shifts it vertically. When we introduce inequalities, we’re essentially defining regions on the graph where certain conditions are met.

Graphing Parabolic Inequalities

Graphing parabolic inequalities is akin to shading regions on a canvas to indicate where certain conditions hold true. Here’s how we can do it:

1. Greater Than (>):For y > a(x – p)^2 + q, we shade the region above the parabola. This indicates that all points above the parabola.

2. Less Than (<): Conversely, for y < a(x – p)^2 + q, we shade the region below the parabola. This indicates that all points below the parabola.

3. Greater Than or Equal To (≥): For y ≥ a(x – p)^2 + q, we include the parabola in the shading, indicating that points on the parabola or above it.

4. Less Than or Equal To (≤): Similarly, for y ≤ a(x – p)^2 + q, we include the parabola in the shading, indicating that points on the parabola or below it.

 

NOTE- To figure out which region you have to shade, we can use (0,0) points in the formula. If when you solve it becomes true then that means you shade that region, and if it doesn’t you shade the opposite region.

NOTE- When ever we have >,< without the equal sign that means we have a dotted line. And whenever we have ≥,≤ that makes just the line without the dots.

Examples

Let’s dive into some examples to solidify our understanding:

1. y > -2(x – 1)^2 + 3:
– We start by graphing the parabola y = -2(x – 1)^2 + 3.

– Next, we shade the region above the parabola to indicate all points where y is greater than -2(x – 1)^2 + 3.

2. y < 4(x + 2)^2 – 1:
– We graph the parabola y = 4(x + 2)^2 – 1.

– Then, we shade the region below the parabola to indicate all points where y is less than 4(x + 2)^2 – 1.

3. y ≥ 1/2(x – 3)^2 + 2:
– We graph the parabola y = (1/2)(x – 3)^2 + 2.

– We shade the region above the parabola and include the parabola itself to show that all points on or above the parabola.

4. y ≤ -3(x + 1)^2 – 4:
– We graph the parabola y = -3(x + 1)^2 – 4.

– We shade the region below the parabola and include the parabola itself to indicate that all points on or below the parabola.

Conclusion

By graphing parabolic inequalities in the form y = a(x – p)^2 + q, we gain valuable insights into the relationships between variables and conditions. Whether it’s shading regions above, below, or including the parabolas themselves, graphing inequalities allows us to visualize solutions and make informed decisions in various mathematical contexts. So, let’s continue to explore the fascinating world of parabolic inequalities and unleash the power of graphing to illuminate our mathematical journey!