Vertex Form and Parabola Basics

Every parabola has an axis of symmetry that divides it into perfect mirror images and passes through its vertex. Quadratic functions can be written in several forms, with the vertex form being particularly useful: f(x) = a$x-h$² + k.

In vertex form, the value of a determines both the parabola's opening direction and stretching. When a > 0, the parabola opens upward; when a < 0, it opens downward. The larger the absolute value of a, the narrower the parabola (|a| > 1 creates a vertical stretch, while 0 < |a| < 1 creates a vertical compression).

The point (h, k) is the vertex of the parabola, representing either its minimum value (when opening upward) or maximum value (when opening downward). The equation x = h describes the axis of symmetry.

💡 Quick Tip: When graphing a quadratic function in vertex form, always start by plotting the vertex - it's the easiest point to identify and serves as your anchor point!

Converting to Standard Form

The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. You can convert from vertex form to standard form by expanding the expression.

To find the vertex from standard form, you can use the formula x = -b/2a to find the x-coordinate, then substitute this value into the original equation to find the y-coordinate. For example, if f(x) = -2x²-12x-14, you'd calculate x = -(-12)/2(-2) = 12/4 = 3.

The x-intercepts are the points where the parabola crosses the x-axis $y = 0$, and the y-intercept is where the parabola crosses the y-axis $x = 0$. The y-intercept equals the c-value in standard form.

🔑 Remember: When a parabola opens upward, the vertex is a minimum point, and the function decreases then increases. When it opens downward, the vertex is a maximum point, and the function increases then decreases.

Parabola Structure and Focus-Directrix Relationship

A parabola has some special geometric properties that define its structure. The focus is a fixed point that lies inside the parabola on the axis of symmetry. The directrix is a fixed line perpendicular to the axis of symmetry.

Every point on the parabola is equidistant from the focus and the directrix - this is actually the definition of a parabola! The vertex sits exactly halfway between the focus and the directrix.

Quadratic functions can be written in three important forms:

• Vertex form: f(x) = a$x-h$² + k, with vertex at (h,k)
• Intercept form: f(x) = a$x-p$$x-q$, where p and q are x-intercepts
• Standard form: f(x) = ax²+bx+c, with x-coordinate of vertex at x = -b/2a

🌟 Insight: Understanding these different forms gives you flexibility when solving problems! Choose the form that makes your specific task easiest.

Horizontal and Vertical Parabolas

Vertical parabolas (opening up or down) have equations in the form y = ax² or more generally y = 4px². The parameter p helps determine the focus and directrix locations.

Horizontal parabolas (opening left or right) have equations in the form x = ay² or more specifically x = 4py². For these parabolas, when p > 0, they open to the right; when p < 0, they open to the left.

For vertical parabolas, the axis of symmetry is a vertical line $x = h$, while for horizontal parabolas, the axis of symmetry is a horizontal line $y = k$. Creating a T-table can help you find points on the parabola to aid in graphing.

🔍 Helpful Hint: Notice how the equations switch between x and y when the parabola changes orientation! This reflects the fundamental difference between vertical and horizontal parabolas.

Focus and Directrix for Vertical Parabolas

For parabolas with a vertical axis of symmetry $x = h$, the equation can be written as y = 1/(4p)$x-h$² + k. This form directly connects to the focus and directrix.

The focus of this parabola is located at the point $h, k+p$ - that's the vertex with p added to the y-coordinate. The directrix is the horizontal line y = k-p, which is exactly the same distance below the vertex as the focus is above it.

When graphing, first identify the vertex (h,k), then locate the focus and directrix using the p value. The axis of symmetry passes vertically through both the vertex and focus. You can create a table of points to help visualize the curve.

📐 Math Insight: The parameter p determines how "shallow" or "steep" your parabola will be. A larger |p| value creates a wider parabola, while a smaller |p| value creates a narrower one.

Focus and Directrix for Horizontal Parabolas

Horizontal parabolas have equations of the form x = 4p$y-k$² + h with the axis of symmetry y = k. Their structure mirrors vertical parabolas but with x and y roles swapped.

The focus of a horizontal parabola sits at $h+p, k$ - that's p units to the right of the vertex when p > 0, or p units to the left when p < 0. The directrix is the vertical line x = h-p, always on the opposite side of the vertex from the focus.

When p > 0, the parabola opens to the right; when p < 0, it opens to the left. You can identify the opening direction quickly by checking the sign of p in the equation.

💡 Visual Tip: Draw a quick sketch showing the vertex, focus, and directrix to help visualize the parabola's orientation and opening. The parabola always opens away from the directrix and toward the focus!

Different Forms for Different Situations

Choosing the right form for a quadratic equation depends on what information you have:

Vertex form $y = a(x-h)² + k$ works best when you know the vertex and one other point. This form makes it easy to see where the highest/lowest point is and the direction the parabola opens.

Intercept form $y = a(x-p)(x-q)$ is ideal when you know the x-intercepts. The zeros of the function are immediately visible as p and q, and the vertex is halfway between them.

Standard form $y = ax² + bx + c$ is useful when you have three points and need to set up a system of equations. The y-intercept is immediately visible as c.

🧩 Strategy Tip: When solving quadratic modeling problems, first identify what information you have available, then choose the form that best fits that information!

Modeling with Quadratics

When modeling real-world situations with quadratic functions, you'll often need to find the equation based on given points. The approach depends on what information you have.

If you know the vertex (h,k) and another point (x,y), use vertex form y = a$x-h$² + k. Substitute the second point to solve for a. For example, with vertex (3,2) and point (13,8), set up 8 = a(13-3)² + 2, then solve to get a = 6/100 or 0.06.

These modeling techniques allow you to create quadratic functions that perfectly fit specific criteria. Once you have your equation, you can analyze key features like maximum/minimum values, intercepts, and other points on the curve.

🔮 Application Note: Quadratic functions model many real-world phenomena like projectile motion, profit optimization, and bridge arches. Mastering these techniques helps you analyze practical situations!