Parabola

A parabola is a type of conic section that forms a U-shaped curve. It is defined as the set of all points that are equidistant from a fixed line (the directrix) and a fixed point (the focus).

Standard Form of a Parabola

The standard form of a parabola with vertex (h, k) and a vertical axis of symmetry is given by the equation:

y = a(x-h)^2 + k

Where (h, k) is the vertex of the parabola and 'a' determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downwards.

Key Features of a Parabola

Vertex: The point (h, k) at which the parabola reaches its minimum or maximum value.
Axis of Symmetry: The vertical line that passes through the vertex of the parabola.
Focus: The fixed point that is equidistant from all points on the parabola.
Directrix: The fixed line that is equidistant from all points on the parabola.
Minimum/Maximum Value: The lowest or highest point of the parabola, located at the vertex.

Graphing a Parabola

To graph a parabola in standard form, start by plotting the vertex (h, k). Then use the value of 'a' to determine the direction and width of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downwards. Use the axis of symmetry to plot additional points on the parabola.

Examples

Let's consider the parabola given by the equation y = 2(x-3)^2 + 1.

Vertex: (3, 1)
Axis of Symmetry: x = 3
Focus and Directrix can be calculated using the formulae:
Focus: (h, k + 1/(4a)) = (3, 1 + 1/(4*2)) = (3, 1.125)
Directrix: y = k - 1/(4a) = 1 - 1/(4*2) = 0.875

Study Guide

When studying parabolas, it is important to understand the standard form of the equation, as well as the key features such as the vertex, axis of symmetry, focus, and directrix. Practice graphing parabolas and identifying these key features from the equation. Additionally, understanding how 'a' affects the parabola's direction and width is crucial for mastering this topic.

Be sure to review the formulas for finding the focus and directrix of a parabola, and practice applying these formulas to different examples.

◂Math Worksheets and Study Guides Seventh Grade. Decimal Operations

The resources above cover the following skills:

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Work flexibly with fractions, decimals, and percents to solve problems.

Develop meaning for integers and represent and compare quantities with them.

Understand meanings of operations and how they relate to one another.

Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

Compute fluently and make reasonable estimates.

Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods.

Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.

Grade 7 Curriculum Focal Points (NCTM)

Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations

Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation.