Vertex Form of Quadratic Equations

This page focuses on the vertex form of quadratic equations and how to interpret it.

The vertex form of a quadratic equation is y = a$x - h$² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful for identifying the axis of symmetry and vertex directly from the equation.

The page provides practice problems for identifying the axis of symmetry and vertex from equations in vertex form. It also includes examples of graphing quadratic functions in vertex form.

Definition: The vertex form of a quadratic equation is y = a$x - h$² + k, where (h, k) is the vertex of the parabola.

Example: For y = -$x - 3$², the axis of symmetry is x = 3, and the vertex is (3, 0).

Highlight: The vertex form is particularly useful for quickly identifying the key features of a quadratic function, such as its vertex and axis of symmetry.

Transformations from the Parent Function

This page explores transformations of quadratic functions from the parent function y = x².

Transformations can include:

• Vertical and horizontal shifts
• Reflections
• Vertical stretches or compressions

The page provides examples of graphing transformed quadratic functions and describing how they compare to the parent function.

Vocabulary: A transformation is a change to the size, shape, or position of a figure.

Example: The function y = $x + 2$² represents a shift of the parent function 2 units to the left.

Highlight: Understanding transformations is crucial for quickly sketching quadratic functions and predicting their behavior based on their equation.

Writing Equations and Putting It All Together

This final page combines all the concepts learned to write equations based on described transformations.

Key points about the vertex form y = a$x - h$² + k:

• h represents the horizontal shift (positive h shifts left, negative h shifts right)
• k represents the vertical shift (positive k shifts up, negative k shifts down)
• If a is negative, the function is reflected over the x-axis
• |a| > 1 represents a vertical stretch, while 0 < |a| < 1 represents a vertical compression

The page includes practice problems for writing equations based on described transformations from the parent function.

Example: A function translated 3 units left and 4 units down can be represented as y = $x + 3$² - 4.

Highlight: Being able to write equations based on described transformations demonstrates a deep understanding of how the coefficients and constants in a quadratic equation affect its graph.

Vocabulary: Vertical stretch occurs when |a| > 1, while vertical compression occurs when 0 < |a| < 1 in the vertex form of a quadratic equation.

Graphing Quadratic Equations

This page introduces the steps to graph a quadratic equation and provides examples for practice.

To graph a quadratic equation:

1. Find the axis of symmetry using the formula x = -b/(2a)
2. Calculate the vertex
3. Create a table of values
4. Plot points and connect them into a smooth parabola

The page includes practice problems for graphing quadratic equations and identifying key features such as the axis of symmetry, vertex, domain, and range.

Example: For y = x² + 2x - 1, the axis of symmetry is x = -1, and the vertex is $-1, -2$.

Vocabulary: Domain refers to all possible x-values, while range refers to all possible y-values of the function.

Highlight: Understanding how to graph quadratic equations is crucial for visualizing their behavior and identifying important characteristics.