Graph 4x2 + y2 = 9. What are its lines of symmetry?
lines are the x,y axes, since
f(x,y) = f(-x,y) = f(x,-y)
also the origin is a point of symmetry since f(x,y) = f(-x,-y)
To find the lines of symmetry of the given equation 4x^2 + y^2 = 9, we can follow these steps:
Step 1: Rewrite the equation in standard form.
The given equation can be rewritten as:
4x^2 + y^2 = 9
Step 2: Divide both sides of the equation by 9 to isolate the squared terms.
Dividing both sides by 9, we get:
(4x^2)/9 + (y^2)/9 = 1
Step 3: Rearrange the equation to obtain the equation of an ellipse.
To make the equation resemble the standard form of an ellipse, we need to make the coefficients of x^2 and y^2 equal to 1. To do this, divide both terms by 9/4:
(x^2)/(9/4) + (y^2)/(9/4) = 1
Simplifying, we get:
(x^2)/(3/2)^2 + (y^2)/(3/2)^2 = 1
Which can be further simplified as:
(x^2)/(3/2)^2 + (y^2)/(3/2)^2 = 1
Comparing with the standard form of an ellipse, we have:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
Where (h,k) represents the center of the ellipse, a represents the semi-major axis, and b represents the semi-minor axis.
Step 4: Identify the semi-major and semi-minor axes.
From the standard form of an ellipse, we can determine that the semi-major axis (a) is equal to the square root of the denominator of the x-term when the equation is in proper form. In this case, a = √(3/2)^2 = 3/2.
Similarly, the semi-minor axis (b) is equal to the square root of the denominator of the y-term, which is also 3/2.
Step 5: Determine the center of the ellipse.
To find the center of the ellipse, we look at the terms inside the parentheses. The center (h, k) is the opposite of these terms. In this case, the center is at (0, 0) since there are no x or y values inside the parentheses.
Step 6: Recall the properties of an ellipse.
An ellipse is symmetric with respect to its center. Therefore, the lines of symmetry of this ellipse are the vertical and horizontal lines passing through the center, which in this case are the y-axis (x = 0) and the x-axis (y = 0), respectively.
So, the lines of symmetry for the ellipse 4x^2 + y^2 = 9 are x = 0 and y = 0.
To find the lines of symmetry of the graph 4x^2 + y^2 = 9, we can determine the shape of the graph by rearranging the equation into a standard form.
Start by dividing both sides of the equation by 9 to get:
4x^2/9 + y^2/9 = 1
Next, let's rewrite the equation to isolate y^2 by subtracting 4x^2/9 from both sides:
y^2/9 = 1 - 4x^2/9
Now multiply both sides of the equation by 9 to eliminate the denominator on the left side:
y^2 = 9 - 4x^2
To further simplify, we can rewrite the equation in terms of y:
y^2 = 9(1 - (4/9)x^2)
Now, since the coefficient of x^2 is negative, we know that the graph will be an ellipse.
The standard form of an ellipse centered at the origin is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse, and a and b are the semi-major and semi-minor axes.
Comparing our equation to the standard form, we can determine that the center of the ellipse is (0,0). And the length of the semi-major axis, a, is equal to the square root of 9, which is 3. The length of the semi-minor axis, b, is also equal to the square root of 9, which is 3.
Now that we know the shape and size of the ellipse, we can identify its lines of symmetry. In an ellipse, the lines of symmetry pass through the center and are parallel to the major and minor axes.
Since our ellipse has equal semi-major and semi-minor axes, the lines of symmetry will be parallel to both the x and y axes. Therefore, the lines of symmetry for the graph 4x^2 + y^2 = 9 are the x-axis (y = 0) and the y-axis (x = 0).