If you draw the graph for a quadratic equation, you can get the shape parabola.
Each quadratic functions will have some characteristics.
They are
Axis of Symmetry :
The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola . The x -coordinate of the vertex is the equation of the axis of symmetry of the parabola.
X and Y Intercepts :
The point at which the parabola cuts the x-axis is known as x-intercept.To find x-intercept we have to put y = 0.
The point at which the parabola cuts the y-axis is known as y-intercept.To find y-intercept we have to put x = 0.
Zeroes :
We can get the zeroes of a quadratic function by applying y = 0. Zeroes of a quadratic function and x-intercepts are same.
Vertex :
The vertex of a parabola is the point where the parabola crosses its axis of symmetry.
The vertex of the parabola is the highest or lowest point also known as maximum value or minimum value of the parabola.
Point Symmetric to Y-Intercept :
The y-intercept (and other points) can be reflected across the axis of symmetry to find other points on the graph of the function.
The points which are having same horizontal distance from the axis is known as symmetric points.
Symmetric points is also known as mirror point.
Example :
Find the equation of axis of symmetry, x and y intercepts, zeroes, vertex and point symmetric to y-intercept. Sketch the graph of the function.
y = x^{2} - 2 x - 1
Solution :
Comparing
y = x^{2} - 2 x - 1
and
y = ax^{2} + bx + c,
we get
a = 1, b = -2 and c = -1
The given parabola is symmetric about y axis.
Because a > 0, the parabola is open upward.
Equation of Axis :
x = -b/2a
x = -(-2)/2(-1)
x = (-2) / (-2)
x = 1
X and Y Intercepts :
Put y = 0. 0 = x^{2} - 2 x - 1 0 = (x - 2)(x + 1) x = -1, 2 x intercepts are -1 and 2 |
Put x = 0. y = 0^{2} - 2(0) - 1 y = -1 y-intercept is -1 |
Zeroes :
Let p(x) = x^{2} - 2 x - 1.
If p(x) = 0, then
x^{2} - 2 x - 1 = 0
Solving the above quadratic equation using quadratic formula, we get
x = (2 ± √8)/2
x = 1 ± √2
x = 1 + √2, x = 1 - √2
x = 2.414, -0.414
Vertex :
Formula to find x-coordinate of the vertex is
x = -b/2a
x = -(-2) / 2(1)
x = 2/2
x = 1
Substitute x = 1 in the given equation function.
y = 1^{2} - 2(1) - 1
y = 1 - 3
y = -2
Vertex of the parabola is (1, -2)
Point Symmetric to Y-Intercept :
The point symmetric to y intercept will have the same horizontal distance from the axis of symmetry.
To find that point we have to substitute y-intercept into the given function.
-1 = x^{2} - 2 x - 1
Add 1 to each side.
0 = x^{2} - 2x
0 = x(x - 2)
x = 0 and x = 2
So, points symmetric to y-intercept are
(0, -1) and (2, -1)
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
You can also visit our following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits