What is the y-intercept of a parabola in Algebra 1?

📚 Understanding the Y-Intercept of a Parabola

The y-intercept of a parabola is the point where the parabola intersects the y-axis. In other words, it's the y-value when $x = 0$. This point is often written as $(0, y)$. Finding the y-intercept is crucial for understanding the behavior and position of a parabola on a graph.

📜 History and Background

The study of parabolas dates back to ancient Greece, with mathematicians like Menaechmus and Apollonius exploring their properties. Parabolas are conic sections, formed when a plane intersects a cone parallel to one of its sides. Their relevance extends to various fields, from optics (reflecting telescopes) to physics (projectile motion).

➗ Key Principles

• 🔍 Standard Form: When a parabola is given in standard form, $y = ax^2 + bx + c$, the y-intercept is simply the constant term, $c$. This is because when $x = 0$, the equation becomes $y = a(0)^2 + b(0) + c$, which simplifies to $y = c$.
• 📈 Vertex Form: If the parabola is given in vertex form, $y = a(x - h)^2 + k$, you can find the y-intercept by substituting $x = 0$ into the equation. This gives $y = a(0 - h)^2 + k = ah^2 + k$. Therefore, the y-intercept is $ah^2 + k$.
• 🌐 Factored Form: If the parabola is given in factored form, $y = a(x - r_1)(x - r_2)$, you can similarly substitute $x = 0$ to find the y-intercept: $y = a(0 - r_1)(0 - r_2) = a(r_1)(r_2)$.
• ✏️ Graphical Method: Visually, the y-intercept is the point where the parabola crosses the y-axis. If you have the graph of the parabola, you can directly read off the y-coordinate of this intersection point.

🧮 Real-World Examples

Let's consider some examples to illustrate how to find the y-intercept:

1. 📝 Example 1: Given the equation $y = 2x^2 + 3x + 5$, the y-intercept is simply $5$. The point is $(0, 5)$.
2. 📐 Example 2: Given the equation $y = -(x - 1)^2 + 4$, substitute $x = 0$: $y = -(0 - 1)^2 + 4 = -1 + 4 = 3$. The y-intercept is $3$. The point is $(0, 3)$.
3. 📊 Example 3: Given the equation $y = (x + 2)(x - 3)$, substitute $x = 0$: $y = (0 + 2)(0 - 3) = (2)(-3) = -6$. The y-intercept is $-6$. The point is $(0, -6)$.

📝 Practice Quiz

Find the y-intercept for each of the following parabolas:

1. ❓$y = x^2 - 4x + 3$
2. ❓$y = 3(x + 1)^2 - 2$
3. ❓$y = -2(x - 2)(x + 1)$

Answers:

1. ✅ $3$
2. ✅ $1$
3. ✅ $8$

💡 Tips and Tricks

• 🧠 Tip 1: Always remember to set $x = 0$ to find the y-intercept.
• 🧮 Tip 2: Be careful with signs when substituting values into equations.
• 📈 Tip 3: Practice with different forms of quadratic equations to become comfortable with identifying the y-intercept quickly.

🔑 Conclusion

Finding the y-intercept of a parabola is a fundamental skill in algebra. Whether the equation is in standard, vertex, or factored form, the method remains the same: set $x = 0$ and solve for $y$. Understanding this concept will greatly enhance your ability to analyze and graph quadratic functions. Keep practicing, and you'll master it in no time!