Hi Dear Jiskha team,
I have Algebra homework and I will need some help.
The homework is following:
Suppose you are using a computer to draw a face on a coordinate plane. The eyes are at (–3, 3) and (3, 3). The tip of the nose is at the origin. To make a smiling mouth, you are going to enter the equation for the graph of a quadratic function, with its appropriate domain.
Post the quadratic function that you would use (in any form) with the domain. Explain how you determined your equation and why you wrote it in the form that you did.
Thanks in advance!
Draw the given items on your calculator or a piece of graph paper.
A quadratic centred on the y-axis has the formula y=x^2+k where k is the displacement from the origin (up:k>0, down:k<0).
The above quadratic is a smiley. If you want a sad face, it's y=-x^2+k.
If you find that the face smiles too much, you can try y=0.5x^2+k, etc.
The domain gives the extent of the mouth, such as from -2 to +2.
Here're some examples where the domain is not specified.
http://img138.imageshack.us/img138/7217/1329904502.png
http://img141.imageshack.us/img141/7217/1329904502.png
To find the equation for the graph of a quadratic function to represent a smiling mouth, we need to consider the shape and properties of a quadratic function.
A quadratic function is typically written in the form of y = ax^2 + bx + c, where "a", "b", and "c" are constants.
In this case, we want a curve that represents a smiling mouth. To achieve this shape, we need a curve that opens upwards (concave up) and intersects the x-axis at two points on both sides of the y-axis (left and right sides of the face).
To create an upward-opening curve, we need to use a positive value for the "a" coefficient in our quadratic function. This will ensure the graph opens upward.
To ensure the curve intersects the x-axis on both sides of the y-axis, we can choose two x-values that correspond to the location of the corners of the mouth.
Since the eyes of the face are located at (-3, 3) and (3, 3), and the tip of the nose is at the origin (0, 0), we can choose the x-values -3 and 3 as the points where the curve intersects the x-axis.
Now let's find the equation:
Since the curve intersects the x-axis at -3 and 3, the factors of the quadratic equation are (x - (-3)) and (x - 3). Simplifying this will give us:
(x + 3) and (x - 3).
Multiplying these factors together will give us the quadratic equation:
y = a(x + 3)(x - 3).
To determine the value of "a" we can consider the y-coordinate of the eyes which is 3. Since the eyes are at the highest point of the smile, this gives us the leading coefficient of the quadratic, so we can set "a" as 3.
Substituting the value of "a" into the equation, we get:
y = 3(x + 3)(x - 3).
Therefore, the quadratic function representing the smiling mouth is y = 3x^2 - 9.
The domain of the function, which represents the set of all possible x-values, can be any real number since the smile can extend to any point on the x-axis.
So, the quadratic function for the graph of a smiling mouth with the appropriate domain is y = 3x^2 - 9.