A parabola is a U-shaped curve that represents quadratic functions. The standard form is y equals a x squared plus b x plus c. The coefficient 'a' is crucial as it determines both the opening direction and the width of the parabola. When 'a' is positive, the parabola opens upward, and when 'a' is negative, it opens downward. Larger absolute values of 'a' make the parabola narrower, while smaller values make it wider.
Now let's analyze the key components of a parabola using the example y equals x squared minus 4x plus 3. The vertex is the highest or lowest point of the parabola. We can find its x-coordinate using the formula x equals negative b over 2a. For our example, x equals 2, giving us the vertex at (2, -1). The axis of symmetry is a vertical line passing through the vertex. The y-intercept occurs where x equals zero, which is the point (0, 3). The x-intercepts, also called roots, are where the parabola crosses the x-axis at (1, 0) and (3, 0).
Parabolas represent quadratic functions, which follow the form f of x equals a x squared plus b x plus c. As functions, parabolas pass the vertical line test, meaning each x-value corresponds to exactly one y-value. The domain of a quadratic function is all real numbers, from negative infinity to positive infinity. However, the range is bounded by the vertex. For example, if we input x equals 2 into our function, we get f of 2 equals negative 1, demonstrating the input-output relationship that defines functions.
The vertex form of a parabola is f of x equals a times x minus h squared plus k, where h and k represent the coordinates of the vertex. This form makes transformations clear. Starting with the parent function f of x equals x squared, we can apply transformations. The parameter h shifts the parabola horizontally, k shifts it vertically, and a controls stretching, compression, and reflection. In our example, f of x equals negative 0.5 times x minus 1 squared plus 3, the vertex moves from (0, 0) to (1, 3), showing a horizontal shift of 1 unit right and vertical shift of 3 units up, with reflection and compression due to a equals negative 0.5.
Parabolas appear frequently in real-world applications. In projectile motion, the path of a thrown ball follows a parabolic trajectory described by h of t equals negative 16 t squared plus v naught t plus h naught. Here we see a ball thrown with initial velocity 64 feet per second from height 80 feet. The maximum height occurs at the vertex at 2 seconds and 144 feet. Parabolas also appear in architecture, like parabolic arches, and in optimization problems where we need to find maximum profit or minimum cost using the vertex of the parabolic function.