A quadratic function is a polynomial function of degree 2. It has the standard form f of x equals a x squared plus b x plus c, where a is not equal to zero. The coefficient a determines the parabola's direction and width, b affects the horizontal position, and c represents the y-intercept. Here are some examples: f of x equals x squared, where a equals 1, b and c equal 0. Another example is f of x equals 2 x squared plus 3 x plus 1, with a equals 2, b equals 3, and c equals 1.
Quadratic functions create parabolic curves when graphed. The shape and direction depend on the coefficient a. When a is positive, the parabola opens upward like a U-shape. When a is negative, it opens downward like an upside-down U. The magnitude of a affects the width: larger values make narrower parabolas, while smaller values create wider ones. Here we see three examples: f of x equals x squared opens upward, f of x equals 2 x squared is narrower, and f of x equals negative x squared plus 4 opens downward.
Let's analyze how each coefficient affects the quadratic function. The coefficient 'a' determines both the opening direction and width of the parabola. Positive values open upward, negative values open downward, and larger absolute values create narrower parabolas. The coefficient 'b' affects the horizontal position and determines the axis of symmetry. The coefficient 'c' represents the y-intercept, where the parabola crosses the y-axis. Watch as we demonstrate these effects by changing each coefficient systematically.
The vertex is the most important point on a parabola - it's either the highest point when the parabola opens downward, or the lowest point when it opens upward. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. To find the vertex, we use the formula x equals negative b divided by 2a for the x-coordinate, then substitute this value back into the function to find the y-coordinate. In this example, the parabola f of x equals x plus 1 squared minus 1 has its vertex at negative 1, negative 1, and the axis of symmetry is the line x equals negative 1.
Quadratic functions have many practical applications in real life. In projectile motion, they describe the path of thrown objects like balls or rockets. In business, they help find optimal pricing for maximum profit. They're also used in area optimization problems. Let's examine a specific example: a ball thrown upward with height function h of t equals negative 16 t squared plus 64 t plus 5. To find the maximum height, we use the vertex formula. The time at maximum height is t equals negative b over 2a, which gives us t equals 2 seconds. At this time, the ball reaches its maximum height of 69 feet before falling back down.