2. Key Concepts

The standard form of a quadratic function is represented as:

$f(x) = ax^2 + bx + c$

A quadratic function is expressed using variables and constants, specifically as a polynomial of degree 2. The symbols 'a', 'b', and 'c' are coefficients, where 'a' must not be zero (a ≠ 0). 'x' is the independent variable that represents the input values of the function. This form is fundamental as it allows us to analyze the function's properties, including its shape and behavior.

Think of the quadratic function as a recipe. The ingredients (a, b, and c) indicate how much of each you need to create a specific dish (the parabola). Just as you can't make a cake without flour, you can't have a quadratic function without the coefficient 'a' being a non-zero number.

• The graph of a quadratic function is called a parabola. • Opens upwards if 𝑎 > 0 • Opens downwards if 𝑎 < 0

When graphed, a quadratic function produces a U-shaped curve known as a parabola. The direction in which this curve opens depends on the coefficient 'a': if 'a' is positive, the parabola opens upward, resembling a smile, whereas if 'a' is negative, it opens downward, like a frown. This characteristic is essential when predicting how the function behaves for different values of x.

Imagine holding a tennis ball. If you throw the ball upwards, it travels in a path similar to an upward-opening parabola until it reaches its peak and then falls back down, showcasing how the shape of the path mirrors the behavior of quadratic functions.

• The vertex is the maximum or minimum point of the parabola. • Formula to find the vertex: $x = -\frac{b}{2a}, \, y = f\left(-\frac{b}{2a}\right)$

The vertex marks where the parabola changes direction; it can be the highest point (if the parabola opens downwards) or the lowest point (if it opens upwards). To locate the vertex quantitatively, we can use the formula for the x-coordinate and evaluate it in the function to find the corresponding y-coordinate. This point is crucial because it often represents the extremum of the function.

Think of a roller coaster. The vertex is the highest or lowest point of the ride, depending on whether it goes up or down first. Just like riders can anticipate the thrill of descending after reaching the top, we can determine critical effects of the quadratic function by finding its vertex.

• A vertical line that passes through the vertex. $x = -\frac{b}{2a}$

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. This line passes through the vertex and can be found using the same formula as for the x-coordinate of the vertex. Understanding the axis of symmetry helps in graphing the function and in analyzing its properties, as any point on one side of the parabola will have a corresponding point on the other side at an equal distance from this line.

Picture a perfectly balanced seesaw. The axis of symmetry acts like the central pivot point of the seesaw, ensuring that both sides are equal. Similarly, the axis creates balance for the parabola, showing symmetry in its shape.

• Set 𝑥 = 0 in the equation: 𝑓(0) = 𝑐

The y-intercept occurs where the graph intersects the y-axis, which happens when the x-value is zero. By substituting '0' in the quadratic function's formula, we can easily find the corresponding y-value, which equals 'c'. Knowing the y-intercept is essential for plotting the graph accurately.

Imagine the y-axis as a starting line in a race. The point where the racers cross this line for the first time represents the y-intercept. It gives us valuable information about the initial conditions before any movement occurs along the x-axis.