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The vertex form of a quadratic function is a special way to express the equation of a parabola. It is written as: \[ y = a(x-h)^2 + k \]where

• \(a\) determines the width and direction of the parabola. If \(a\) is positive, the parabola opens upwards, if negative, it opens downwards.
• \(h\) and \(k\) are the coordinates of the vertex of the parabola.

The vertex is the peak or the lowest point of the parabola, and it can be easily read directly from the equation. In the given equation, \( y = -2(x+3)^2 + 2 \), the vertex is \((-3, 2)\). This form is particularly useful as it provides clear information about the vertex and simplifies the process of graphing the parabola.

The axis of symmetry is an imaginary vertical line that divides the parabola into two mirror-image halves. Every point on one side of the axis has a corresponding point on the opposite side. For the vertex form equation \( y = a(x-h)^2 + k \), the axis of symmetry has the equation: \[ x = h \]This means that the axis of symmetry is always a vertical line passing through the vertex's x-coordinate. For our example equation, \( y = -2(x+3)^2 + 2 \), the axis of symmetry is: \[ x = -3 \]This simplifies the graphing process, as you can ensure both sides of the parabola are symmetrical about this line.

The domain and range of a quadratic function describe the set of possible input (x-values) and output (y-values), respectively. **Domain**For any quadratic function, including our example \( y = -2(x+3)^2 + 2 \), the domain is all real numbers. This is because parabolas extend infinitely in both directions along the x-axis. Thus, we write the domain as: \[ x \in (-\infty, \infty) \]**Range**The range, however, depends on the direction the parabola opens. Since our parabola opens downward (indicated by the negative \( a\)), it has a maximum value at the vertex, which is the y-coordinate of the vertex. Therefore, the range for our parabola is all values less than or equal to 2: \[ y \in (-\infty, 2] \]This means the function produces values from negative infinity up to the vertex height.

A quadratic function is a polynomial function of degree 2, characterized by the form: \[ y = ax^2 + bx + c \]However, rearranging into vertex form by completing the square can provide insights that are more intuitive for graphing, as seen in our example problem.

• **Standard Form:** Gives immediate information about the parabola's y-intercept (\( c \)) and can be transformed into vertex form to find the vertex easily.
• **Vertex Form:** Immediately tells us the vertex and can also be used to determine the direction and stretch of the parabola.

Quadratic functions graph as a symmetrical curve called a parabola. Depending on the sign of \( a\) in vertex form, the parabola either opens upwards (positive \( a\)) or downwards (negative \( a\)). This fundamental understanding aids in accurate hand graphing and verifying with technology, as seen in our example of \( y = -2(x+3)^2 + 2 \). The technique allows identification of critical features like vertex and axis without complex calculations.