Math 1122 Lecture 4 Notes

Math 1122 Lecture 4 Notes
Review of Function Evaluation and Expansion

  • Given function:

    f(x)=2x2+5f(x) = 2x^2 + 5

  • Objective: Evaluate and expand f(y−4)f(y - 4).

  • Step-by-Step Calculation:

    1. f(y−4)=2(y−4)2+5f(y - 4) = 2(y - 4)^2 + 5

    2. Expand (y−4)2(y - 4)^2:

    • Using the formula:

      (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2,

      where a=ya = y and b=4b = 4

    • Therefore:

      (y−4)2=y2−8y+16(y - 4)^2 = y^2 - 8y + 16

    1. Substitute back into the function:

      f(y−4)=2(y2−8y+16)+5f(y - 4) = 2(y^2 - 8y + 16) + 5

    2. Distributing the 2:

      =2y2−16y+32+5= 2y^2 - 16y + 32 + 5

    3. Combine like terms:

      =2y2−16y+37= 2y^2 - 16y + 37

  • Short Way for Evaluation:

    Final result:

    f(y−4)=2y2−16y+37f(y - 4) = 2y^2 - 16y + 37

  • Common Pitfall:

    • Incorrect evaluation such as:

      f(y−4)</p><p>eq2(y2+16)+5=2y2+37f(y - 4)</p><p>eq 2(y^2 + 16) + 5 = 2y^2 + 37

Introduction to New Material: Geometry

  • Defining Geometry:

    • Why is it in 2D?

    • A plane determined by 3 points defines the concept of two-dimensional space.

  • Simplest 2D Shape: Triangle

    • Consider an arbitrary triangle with sides labeled as aa, bb, and cc.

Triangle Properties

  • Triangle Inequality Theorem:

    • The sums of any two sides must be greater than the third side:

    1. a+b>ca + b > c

    2. a+c>ba + c > b

    3. b+c>ab + c > a

Perimeter and Area of Triangles

  • Perimeter:

    • The perimeter of any triangle is given by the sum of its three sides:

      P=a+b+cP = a + b + c

  • Area:

    • The formula for the area of a triangle is:

      Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

    • Here, the 'base' is any side of the triangle, and the 'height' is the perpendicular distance from the opposite vertex to that base.

Special Cases: Right Triangles

  • Definition of Right Triangle:

    • A triangle in which one of the angles measures 90 degrees.

  • Pythagorean Theorem:

    • For right triangles, the relationship between the sides is given by:

      a2+b2=c2a^2 + b^2 = c^2

      where cc is the hypotenuse.

  • Area of Right Triangle:

    • Using the legs as base and height, the area is:

      Area=12ab\text{Area} = \frac{1}{2}ab, where aa and bb are the lengths of the two legs.

Examples of Right Triangles

  • Example 1:

    • Given sides 5, 12 (legs), and 13 (hypotenuse)

    • Verify if:

      52+122=1325^2 + 12^2 = 13^2

    • Calculation:

      25+144=16925 + 144 = 169

      This confirms a right triangle.

Generalizing Triangular Properties

  • Understanding circles and right triangles leads to a general comprehension of all triangles.

  • Goal:

    • To understand all geometry and derive relationships through trigonometry:

    • Trigonometry is the study of the relationships between the angles and sides of triangles and is fundamental for understanding broader geometric principles.