Guide :

# A cubic function f:x = ax^3 +b , where a and b are real numbers and a not equal to 0.

a) Investigate the shape of the graph of the cubic function for various value of a for the case b=0 . Identify the point of inflexion. b) Investigate the shape of the graph if both a  and b   have the same sign and if a  andb  have different signs. Identify the point of inflexion in each case.  c) Investigate the points of intersection of the graphs of f and its tangents. What can u say abt the number of point of intersection.

## Research, Knowledge and Information :

### Cubic function - Wikipedia

The critical points of a cubic function f defined by f(x) = ax 3 + bx 2 ... both Δ and Δ 0 are equal to 0 and is also ... q are real numbers and q 2 / 4 + p 3 / 27 ...
Read More At : en.wikipedia.org...

### Graphing Cubic Functions - Free Mathematics Tutorials ...

Graphing Cubic Functions. ... c and d are real numbers and a is not equal to 0. ... Example 3: f is a cubic function given by f (x) ...
Read More At : www.analyzemath.com...

### Cubic equations - mathcentre.ac.uk - Mathematics resources

ax3 +bx2 +cx+d = 0 where a 6= 0 All cubic equations have either one real ... in x3 or it would not be cubic (and so a 6= 0 ... (x2 +ax+b) = 0 where a and b are numbers.
Read More At : www.mathcentre.ac.uk...

### AmericanoMath - Cubic Functions

Applications of the function to real-world situations; ... b, h and k affect the cubic functions graph. Let's set the cubic function as f(x) = 1/1*(x – 0)^3 + 0 ﻿.
Read More At : americanomath.wikispaces.com...

### Edurite.com - How to Solve Ax3 Bx2 Cx D 0 | Solve for X

... of the equation is 3.Where the value of a,b,c,d are real numbers which ... a,b,c,d such that f(x)=ax^3+bx^2 ... x in the cubic function: ax^3 + bx^2 + cx + d = 0.
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### 1.6 a library of parent functions - UTEP - Academics Portal Index

1.6 A LIBRARY OF PARENT FUNCTIONS ... The graph of the linear function f (x) = ax + b is a line with ... real numbers. • The function is even. 12
Read More At : academics.utep.edu...

### Algebra 2 Unit F Flashcards | Quizlet

Algebra 2 Unit F. STUDY. PLAY. ... a function of the form y=ax^b, ... where a, b & c are real numbers and a is not equal to zero. quartic function.
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### Algebra 2 Honors Flashcards | Quizlet

Algebra 2 Honors. STUDY. PLAY. ... where a,b,and c are real numbers and a ≠ 0. cubic function. a function to the 3d power f(x) ax^3 + bx^2 +cx +d, ...
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## Suggested Questions And Answer :

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### is x^4-x^2+3x-7 a polynomial function

yes it is a polynomial function. What is a polynomial? hmm well its a function that must satisfy the following conditions: 1.) n is a non negative number, meaning its equal to 0 or a positive number. n is the power, in that function the powers, are 4,2 and 1 they all satisfy this. 2.) The coefficients are real numbers. The coefficients here are  1, -1, 3 these are all real numbers so this is satisfied. 3.) the degree of the function is not 0. Degree of the function is the higest power there, so the degree of that function is 4. This is satisfied.
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### polynomial function, i need a definition

Polynomial functions are functions that have this form: f(x) = anxn+ an-1xn-1+ ... + a1x + a0 The value of n must be an non-negative integer. That is, it must be whole number; it is equal to zero or a positive integer. The coefficients, as they are called, are an, an-1,..., a1, a0. These are real numbers. The degree of the polynomial function is the highest value for nwhere an is not equal to 0.     http://zonalandeducation.com/mmts/functionInstitute/polynomialFunctions/definition/definition.html
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### When is the function x^x increasing and decreasing?

x^x = exp (x ln x)   we can say that ln is increasing and x is increasing and exp is increasing so x ln x is increasing and also exp(x ln x) is increasing by composition but for the domain it is not all real numbers except zero  it is all positive real numbers except zero and for negative integers for negative numbers it is defined for some rational n/p  eg  (−1/3)^( −1/3) is not the same as (−2/6)^(−2/6). so to avoid problem it is defined for all positive real numbers except zero
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### Polynomial Functions

Polynomial functions are functions that have this form: f(x) = anxn + an-1xn-1 + ... + a1x + a0 The value of n must be an non-negative integer. That is, it must be whole number; it is equal to zero or a positive integer. The coefficients, as they are called, are an, an-1,..., a1, a0. These are real numbers. The degree of the polynomial function is the highest value for n where an is not equal to 0. So, the degree of
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### What's a function of function?

If you just mean the purpose of a function then see immediately below. If you mean f of g where f and g are two functions, go to the end. The purpose of a function is like giving someone a to-do list. A function (normally shown as f(x)= meaning "function of x", or y=) will usually contain one variable, usually represented by x. Some functions may have more than one variable. If x isn't the variable, it will usually be a different letter like t or a or any letter. Let's say it is x. This variable will appear in a formula with numbers. The formula is the function and it contains instructions in symbols telling you what to do with the variable. For example, multiply the variable by 2 then add 3 and divide the result by 4. This would be written f(x)=(2x+3)/4 or y=(2x+3)/4. The equals sign means that the function is defined as the expression on the right. Just like someone might say to you, think of a number (that's x), but don't tell me what it is, then double it and add 3 and divide the result by 4. Functions can be plotted on a graph. The idea of this is that on the horizontal axis (x axis) you have markers for 0, 1, 2, etc. On the vertical axis (y or f(x)) you also have markers. The graph is usually a continuous line, and every point on the line is made by putting a different value for x and marking the point (on a rectangular grid f(x) units vertically and x units horizontally) for the result of working out the value of the function for different values of x. The points make up the graph. You can use the graph to find the value of x for any value of the function, and the value of the function for any value of x, provided the graph extends far enough. This is just a brief example of the purpose of a function. A practical example is the conversion of temperature in degrees Fahrenheit (F) to degrees Celsius: C=5(F-32)/9. This function tells you what do with the value of the temperature. You could plot this as a straight line graph and you can read off degrees C for any temperature in degrees F, and vice versa. Another example is d=30t where d=distance, speed=30mph, t=time. The function is 30t, and from it you can work out the distance a car moves in a particular time t when its speed is 30mph. Function of a function: this simply means you work out the value of applying one function and then feed this answer as the variable into the other function. An example could be that one function is used to find out how many miles a car travelling at an average speed of 30mph in a given time. The second function could be to find out how much fuel is used to travel that distance. So g(t)=30t is the first function. The second function is h(d)=d/24 where fuel consumption is 24 miles per gallon. h of g is h(g(t))=30t/24=5t/4.
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### Finding the domain of a function

When you're finding the domain of the function, you have to find places where f(x) is undefined. If there are no points where f(x) = undefined, then the domain is all real numbers. There are only a few cases where the domain would be different: -log(x) has no points on or below zero, so to find the domain of log(stuff), then stuff > 0 -f(x)/g(x) (a rational equation) only has undefined points where g(x) = 0 because anything/0 is undefined. So you set the bottom ( g(x) ) not equal to zero and that's your domain -with square root equations, it can't have any points where x is less than zero, so you take the stuff under the square root and set it greater than or equal to 0. Hope this helped :D
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### when is the balloon exactly 980m above the ground

When h(t)=980, -2t^3+3t^2+149t-570=0. The factors of 570=2*3*5*19. From these we can make several composite numbers that could be factors if it is assumed that there are rational zeroes. We need three zeroes or roots and the possible factors are (2,3,95), (2,15,19), (6,5,19), (3,10,19), (2,5,57), (3,5,38). The factors of 2 (t^3 coefficient) are (1,2) only. By testing each factor by dividing the cubic expression one at a time by the factors, both positive and negative values, using, say, synthetic division, we can reduce the cubic to a quadratic. When we do this, we discover that 5 is a zero, and when we divide the expression by t-5 we get -2t^2-7t-114 which further factorises into -(t-6)(2t+19). So the zeroes are 5, 6, -19/2. We can reject the last one, because it's negative and out of the given range (0 to 10). Therefore t=5 or 6 seconds are valid solutions. The balloon reaches 980m at 5 seconds then continues to rise then drops to 980m at 6 seconds.  Check: -250+75+745+410=980=-432+108+894+410.
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### how do i graph |x+2| less than or equal to 3 graph the solution set assuming that the domain is

When x<-2 |x+2| becomes 2-x; when x>-2 |x+2| becomes x+2. So x=-2 is a point to note on a graph. The graph of 2-x has negative slope (\) while x+2 has positive slope. So the graph resembles a V shape with the point of the V at (-2,0), and a y intercept at 2, when x=0. But the limits |x+2|<3 restrict the graph. Draw a line parallel to the x axis at y=3. This line reduces the V to an isosceles triangle with vertices at (-5,3), (1,3) and (-2,0). The domain of x is -5 Read More: ...

### complex, rational and real roots

The quintic function should have 5 roots.  The changes of sign (through Descartes) tell us the maximum number of positive roots. Since there are two changes of sign there is a maximum of 2 positive roots. To find the number of negative roots we negate the terms with odd powers and check for sign changes: that means that there is at most one negative root, because there's only one sign change between the first and second terms. Complex roots always come in matching or complementary pairs, so that means in this case 2 or 4.  Put x=-1: function is positive; for x=-2 function is negative, so there is a real root between -1 and -2, because the x axis must be intercepted between -1 and -2. This fulfils the maximum for negative roots. That leaves 4 more roots. They could be all complex; there could be two complex and two positive roots. Therefore there are at least two complex roots. To go deeper we can look at calculus and a graph of the function: The gradient of the function is 10x^4-5. When this is zero there is a turning point: x^4=1/2. If we differentiate again we get 40x^3. When x is negative this value is negative so the turning point is a maximum when we take the negative fourth root of 1/2; at the positive fourth root of 1/2 the turning point is a minimum, and these are irrational numbers. The value of the function at these turning points is positive. The fourth root of 1/2 is about 0.84. and once the graph has crossed the x axis between -1 and -2, it stays positive, so all other roots must be complex. The function is 12 when x=0, so we can now see the behaviour: from negative values of x, the function intersects the x axis between -1 and -2 (real root); at x=4th root of 1/2 (-0.84) it reaches a local maximum, intercepts the vertical axis at 12 until it reaches 4th root of 1/2 (0.84) and a local minimum, after which it ascends rapidly at a steep gradient.  [Incidentally, one way to find the real root is to rearrange the equation: x^5=2.5x-6=-6(1-5x/12); x=-(6(1-5x/12))^(1/5)=-6^(1/5)(1-5x/12)^(1/5) We can now use an iterative process to find x. We start with x=0, so x0, the first approximation of the solution, is the fifth root of 6 negated=-1.430969 approx. To find the next solution x1, we put x=x0 and repeat the process to get -1.571266. We keep repeating the process until we get the accuracy we need, or the calculator reaches a fixed value. After just a few iterations, my calculator gave me -1.583590704.]
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