2nd Periodical Test Study Notes

2nd Periodical Test Study Notes

Polynomial Functions

Definition and Examples

• Polynomial Function: A function that can be expressed in the form $f(x) = a<em>nx^n + a</em>{n-1}x^{n-1} + … + a<em>1x + a</em>0$ where\

  • Each coefficient $a_n$ is a real number\

  • The degree $n$ is a non-negative integer.

• Examples of polynomial functions:

  • A. $f(x) = 5x^2 + 2x + 3$ (Polynomial since all powers of $x$ are whole numbers)

  • B. $f(x) = 4x^{1/2} - 3x$ (Not a polynomial because of fractional power)

  • C. $f(x) = x^2 + ext{√}2x - 5$ (Not a polynomial due to $ext{√}2x$)

  • D. $f(x) = 5x - 3x^5$ (Polynomial)

Standard Form of a Polynomial

• A polynomial in standard form is ordered by descending power of $x$.

• For example, given $f(x) = 4 + 3x^3 + 5x + 2$, the correct standard form is:

  • B. $f(x) = 3x^3 + 5x + 6$

Polynomial Degree and Turning Points

• Degree of a Polynomial: The highest power of $x$ in the polynomial.

• Each polynomial of degree $n$ can have at most $n$ turning points. Thus:

  • A. At most $n$

  • B. At most $n - 1$ (Correct answer is A)

Calculation of Degree

• To find the degree:

  • Example: For $f(x) = 2(x + 2)^2(x - 1)^4$

  • The degree is calculated as follows:

  • Degree of $(x + 2)^2$ is 2

  • Degree of $(x - 1)^4$ is 4

  • Total degree = $2 + 4 = 6$

• Therefore, the answer is C. 6.

y-Intercept of a Polynomial Function

• The y-intercept indicates where the graph crosses the y-axis (where $x=0$).

• Example: For $f(x) = 5x^3 + 2x^2 - x + 6$,

  • The y-intercept is at (0,6), thus the answer is C. (0,6).

Zeros of a Polynomial Function

• If $f(x) = (x + 2)(x + 1)$,

• Then the zero of the function occurs at $x=-2$.

• From the options, one is incorrect:

  • A. $-2$ is indeed a zero (Correct)

  • D. The graph at $(-2, 0)$ is tangent to x-axis (False)

Leading Coefficients

• The leading coefficient is the coefficient of the term with the highest degree.

• For $f(x) = 3x^2 + 7x + 4$, the leading coefficient is $3$, hence:

  • Thus the answer is C: 3.

Volume of a Box

• Dimensions of a box are given by: $ext{Volume} = (2x+1)(x-1)(x)$.

• To find the volume, expansion is required:

  • Therefore, the correct volume function is B: $V(x) = (2x + 1)(x-1)(x)$.

Value of x for a Volume of 10 Cubic Feet

• Finding dimensions of a box that states volume equals 10 cubic feet requires solving:

  • Given volume function, find $x$ such that volume = 10.$</p></li></ul></li><li><p>The options for x: A. 1, B. 2, C. 3, D. 4</p></li></ul><h5>Demographic Predictions for Population</h5><ul><li><p>The population model is provided by$P = 64 - 53 + 200 + 12000$.</p></li><li><p>We need to calculate the population$P$at$t = 2$years from now.</p></li><li><p>Options given are A. 12 456, B. 124, C. 1 245 6000, D. 12 456 000.</p></li></ul><h4>Profit Function for Car Manufacturer</h4><ul><li><p>The profit equation is given by$P(x) = 0.125x^4 - 5$.</p></li><li><p>Calculate the profit when$x = 100$.</p></li><li><p>Choose from:</p><ul><li><p>A. Php 12,509.50, B. Php 125,095.00, C. Php 1,250,950.00, D. Php 12,500,095.00.</p></li></ul></li></ul><h4>Geometry: Circles and Angles</h4><h5>Definitions</h5><ul><li><p><strong>Circle</strong>: Set of all points equidistant from a fixed center point.</p></li><li><p><strong>Inscribed Angle</strong>: An angle formed by two chords in a circle which meets at an endpoint on the circle.</p></li><li><p><strong>Central Angle</strong>: An angle whose vertex is at the center of the circle.</p></li></ul><h5>Central Angle Measures</h5><ul><li><p>The sum of the measures of all the central angles of a circle is always:</p><ul><li><p>D. 360°</p></li></ul></li></ul><h5>Arc Length Calculation</h5><ul><li><p>Arc of a circle can be calculated using:<br>$ ext{Arc Length} = rac{ heta}{360} * 2 ext{πr}$where$ heta$is the angle in degrees.</p></li><li><p>Given arc measures 30°, with radius 4 cm:</p><ul><li><p>Correct answer is approximately 2.09 cm (A).</p></li></ul></li></ul><h5>Tangent Lines and Secants</h5><ul><li><p><strong>Tangent Line</strong>: A line that touches a circle at exactly one point.</p><ul><li><p>Correct definition is C.</p></li></ul></li></ul><h3>Coordinate Geometry</h3><h5>Distance Between Two Points</h5><ul><li><p>The distance formula between two points is given as:<br>$d = ext{√}((x2 - x1)² + (y2 - y1)² )$</p></li></ul><h5>Distance Example</h5><ul><li><p>For points M(-6, -8) and N(0, 0), calculate:</p><ul><li><p>Correct answer is B. 12 (after calculating).</p></li></ul></li></ul><h5>Circle Equation</h5><ul><li><p>The equation of a circle with center (h,k) and radius r:</p><ul><li><p>$ (x - h)² + (y - k)² = r² $$ (correct form D).

Vertex of a Rectangle

• If vertices are (0,0), (0,5), and (-7,0), the fourth vertex is: B. (-7,5).

Triangle Formed by Points

• Vertices provided are (2,5), (2,-3), and (10,-3), and the type of triangle is:

  • Answer is C. Isosceles Right Triangle.