User:

Documento sin título
 Pre-algebra Arithmetics Integers Divisibility Decimals Fractions Exponents Percentages Proportional reasoning Radical expressions Graphs Algebra Monomials Polynomials Factoring Linear Equations Graphs of linear equations Rectangular Coordinate System Midpoint Formula Definition of Slope Positive and negative slope Determine the slope of a line Equations of lines Equation of lines (from graph) Applications of linear equations Inequalities Quadratic equations Graphs of quadratic equations Absolute Value Radical expressions Exponential equations Logarithmic equations System of equations Graphs and functions Plotting points and naming quadrants Interpreting Graphs Relations and Functions Function Notation Writing a Linear Equation from a Table Writing a Linear Equation to describe a Graph Direct Variation Indirect Variation Domain and range Sequences and series Matrices Inverse of a matrix Determinants Inner product Geometry Triangles Polygons 2-D Shapes 3-D Shapes Areas Volume Pythagorean Theorem Angles Building Blocks Geometry Transformations Parallel, coincident and intersepting lines Distances in the plane Lines in space Plane in space Angles in the space Distances in the space Similarity Precalculus Sequences and series Graphs Graphs Definition of slope Positive or negative slope Determine the slope of a line Equation of a line (slope-intercept form) Equation of a line (point slope form) Equation of a line from graph Domain and range Quadratic function Limits (approaches a constant) Limits (approaches infinity) Asymptotes Continuity and discontinuities Parallel, coincident and intersepting lines Introduction to Functions Limits Continuity Asymptotes Trigonometry Trigonometric ratios The reciprocal trigonometric ratios Trigonometric ratios of related angles Trigonometric identities Solving right angles Law of sines Law of cosines Domain of trigonometric functions Statistics Mean Median Mode Quartiles Deciles Percentiles Mean deviation Variance Standard Deviation Coefficient of variation Skewness kurtosis Frequency distribution Graphing statistics & Data Factorial Variations without repetition Variations with repetition Permutations without repetition Permutation with repetition Circular permutation Binomial coefficient Combinations without repetition Combinations with repetition

Vertex and Intercepts od a Quadratic Graph
Vertex

The quadratic equation has an extreme value located at the vertex of the graph. This vertex point is either the highest point on the graph of the equation or the lowest point:

Finding the vertex of a parabola without graphing
The location of the coordinates of the vertex of the graph of y=ax2+bx+c depends only on the coefficients of a and b:

Coordinates of the vertex: $(\frac{-b}{2a},f(\frac{-b}{2a}))$ (the y-value is found by substituting the x-value)

Find the vertex of f(x)=x2+6x+8
1. Find the x-coordinate of the vertex by using the formula $x=\frac{-b}{2a}$, where a=1 and b=6:
$x=\frac{-6}{2(1)}=-3$

2. Find the y-coordinate of the vertex by evaluating f(-3)

f(-3)=(-3)2+6(-3)+8=9-18+8=-1

Hence, the vertex is at (-3,-1)

Intercepts

The intercepts of a parabola are where the curve or graph of the function crosses the axis.

Finding the intercepts of a parabola without graphing.
• To solve for the x-intercepts of a quadratic function, you set f(x) equal to 0 (this is the same as setting y equal to 0)and then solve for x.
• To solve for the y-intercept of a quadratic function, you set x equal to 0 and solve for f(x). A function has only one y-intercept.

Find the x-intercept(s) and the y-intercept(s) of the quadratic equation y=x2-4

Step 1. First, find the y-intercept. Since the y-intercept is the point where the graph crosses the y-axis, the value for x at this point is zero. Because we know that x=0, plug 0 in for x in the equation and solve for x.

y=x2-4
y=02-4
y=0-4
y=-4
Therefore, the y-intercept is (0,-4)

Step 2. Next, find the x-intercept(s). The x-intercept is the point, or points in this case, where the graph crosses the x-axis. In this case, we plug 0 in for y because y is always zero along the x-axis. Solve for x.

y=x2-4
0=x2-4
0+4=x2-4+4
4=x2
$\sqrt{4}=x$
So x=-2 and x=2
The x-intercepts are (-2,0) and (2,0)

Step 3. To verify that the intercepts are correct, graph the equation on the coordinate plane:

Find the vertex and the intercepts of the quadratic equation: f(x)=x2-x

 x-intercepts y-intercept Vertex ( , ) ( , ) ( , ) ( , )