How do you sketch curves like x to the power of 4 or x to the power of 6 ?
What are the steps involved ?
Could someone please help thankyou!
it will depend what other terms are along with the x^4 term
in general for y = ax^4 + bx^3 +cx^2 + dx + e , the most dramatic graph would be in the shape of a "W", with a maximum possibility of 4 x-intercepts.
in its simplest form of y = x^4, it will resemble a very fast-rising 'parabola'.
I stress that it looks like a parabola, but does not have any of the parabolic qualities.
for y = ax^6 + bx^5 ....
the basic shape would be a double "W" with a maximum of 6 x-intercepts possible.
Your best bet is to make a table of values of x and y. You might want to scale the y axis different from the x axis because the y's tends to become large in a hurry.
I don't really know how much you know about mathematics, but I'll give you some basic steps in sketching a function.
1) find the domain of the function. You can't draw the curves in places where the function can't go.
(For polynomials of x, the function reaches from-infinity untill +infinity)
2) Solve the function. These are the places where your function crosses the x-axis of your graph.
(if you should have any problem solving polynomials of higher degrees, please feel free to ask).
3) Check which value you get for x=0 (this is the place where your curve crosses the y-axis)
4) Next, take the first derivative of the function and solve it. When you solve the first derivative of your function you find the places where your function reaches a maximum or minimum (you can find out if it's a maximum or minimum by making a sign table of the first derivative, if the graph is rising, the first derivative is positive, if the graph is descending, the first derivative is negative)
5)Calculate the second derivative of the function and solve it. The second derivative describes the curvature of your graph. I'll refer to this picture:
Search Google for pictures of "second derivative" and look at the first picture that pops up titled rulers.gif.
In the points P and T, the graph has a negative curvature (negative second derivative) and in R it has a positive curvature (positive second derivative). When the graph flips from positive to negative curvature (or vice versa) it has a second derivative that equals 0.
So by solving the second derivative and making a sign table of it, you can find the points where the graph changes curvature.
6) When you have found all the points I previously discusses, you put them on a graph and connect them with a fluid line. You now should have a graph that is very close to the graph of the function.
To sketch curves like x to the power of 4 or x to the power of 6, you can follow these steps:
1. Determine the domain: Identify the range of values that x can take. For example, if there are no restrictions mentioned, assume x can take any real value.
2. Calculate some points: Choose values of x and calculate the corresponding values of y (which is x^4 or x^6). It's helpful to choose both positive and negative values of x. For example, you can pick x = -2, -1, 0, 1, 2 and calculate their corresponding y values.
3. Plot the points on a graph: Draw a coordinate system with x and y axes. Plot the points you calculated in the previous step on this graph.
4. Connect the points: Once you have a few points plotted, you can start connecting them to get an idea of the shape of the curve. The more points you have, the better the curve representation will be.
5. Examine the behavior as x approaches infinity or negative infinity: Determine how the curve behaves for very large values of x or very small (negative) values of x. This will help you understand any asymptotic behavior or end behavior of the curve.
6. Smooth out the curve: Once you have connected the points, check if there are any smooth curves, sharp turns, or inflection points. Adjust the curve if needed to achieve a smooth transition between points.
7. Label the axes: Finally, label the x and y axes to indicate the variables being represented.
Remember that these steps provide a general guideline. The more points you calculate and plot, the more accurate your sketch will be. Additionally, using graphing software or online graphing tools can help visualize the curve more accurately.