Question

List the intercepts and test for symmetry. $$ 4 x^{2}+y^{2}=4 $$

Short answer

The x-intercepts are (1, 0) and (-1, 0); the y-intercepts are (0, 2) and (0, -2). The equation is symmetric about the x-axis, y-axis, and the origin.

Step-by-step solution

Find the x-intercepts

To find the x-intercepts, set y to 0 and solve for x. Substitute y = 0 into the equation: \[ 4x^2 + (0)^2 = 4 \]Simplify and solve for x: \[ 4x^2 = 4 \ x^2 = 1 \ x = \pm 1 \]Thus, the x-intercepts are (1, 0) and (-1, 0).

Find the y-intercepts

To find the y-intercepts, set x to 0 and solve for y. Substitute x = 0 into the equation: \[ 4(0)^2 + y^2 = 4 \]Simplify and solve for y: \[ y^2 = 4 \ y = \pm 2 \]Thus, the y-intercepts are (0, 2) and (0, -2).

Test for symmetry

To test for symmetry, check the following: 1. **Symmetry about the y-axis: ** Replace x with -x in the equation and check if the equation remains unchanged:\[ 4(-x)^2 + y^2 = 4 \ 4x^2 + y^2 = 4 \]The equation remains unchanged, so it is symmetric about the y-axis.2. **Symmetry about the x-axis: ** Replace y with -y in the equation and check if the equation remains unchanged:\[ 4x^2 + (-y)^2 = 4 \ 4x^2 + y^2 = 4 \]The equation remains unchanged, so it is symmetric about the x-axis.3. **Symmetry about the origin: ** Replace x with -x and y with -y in the equation and check if the equation remains unchanged:\[ 4(-x)^2 + (-y)^2 = 4 \ 4x^2 + y^2 = 4 \]The equation remains unchanged, so it is symmetric about the origin.

Key concepts

x-intercepts

Finding the x-intercepts of an equation means determining where the graph intersects the x-axis. To do this, we set the y-value to zero, because all points on the x-axis have y = 0.

Example: We will use the equation \[ 4x^2 + y^2 = 4 \]
To find the x-intercepts, substitute y = 0 into the equation. We get:
\[ 4x^2 + (0)^2 = 4 \]
Simplifying, this becomes:
\[ 4x^2 = 4 \]
Dividing both sides by 4, we have:
\[ x^2 = 1 \]
Solving for x, we obtain:
\[ x = \text{±} 1 \]
Thus, the x-intercepts are (1, 0) and (-1, 0). This means the graph touches the x-axis at these points.

y-intercepts

Finding y-intercepts involves determining where the graph crosses the y-axis. For this, we set the x-value to zero because all points on the y-axis have x = 0.

Example: Using the same equation, we get:
\[ 4x^2 + y^2 = 4 \]
Substitute x = 0 into the equation:
\[ 4(0)^2 + y^2 = 4 \]
Simplifying, this becomes:
\[ y^2 = 4 \]
Taking the square root of both sides, we have:
\[ y = \text{±} 2 \]
Therefore, the y-intercepts are (0, 2) and (0, -2). So, the graph crosses the y-axis at these points.

symmetry about the axes

Testing for symmetry in a graph helps us understand its overall shape and behavior. We will check for symmetry about the y-axis, x-axis, and the origin.

• Symmetry about the y-axis: Replace x with -x in the equation and check if it remains unchanged. For equation \[ 4(-x)^2 + y^2 = 4 \]
  Simplifying, we get:
  \[ 4x^2 + y^2 = 4 \]
  The equation is unchanged, indicating symmetry about the y-axis.
  
  
• Symmetry about the x-axis: Replace y with -y in the equation and see if it remains the same. For equation \[ 4x^2 + (-y)^2 = 4 \]
  This simplifies to:
  \[ 4x^2 + y^2 = 4 \]
  Again, the equation remains unchanged, showing symmetry about the x-axis.
  
  
• Symmetry about the origin: Replace x with -x and y with -y in the equation. For equation \[ 4(-x)^2 + (-y)^2 = 4 \]
  Simplifying, we have:
  \[ 4x^2 + y^2 = 4 \] The unchanged equation shows symmetry about the origin.