Question

The intercepts of the equation \(x^{2}+4 y^{2}=16\) are ______.

Short answer

(4,0),(-4,0),(0,2),(0,-2)

Step-by-step solution

Understand the Intercepts

Intercepts are the points where the graph of the equation crosses the axes. To find where the graph crosses the x-axis, set y = 0 and solve for x. To find where it crosses the y-axis, set x = 0 and solve for y.

Find x-intercepts

Set y = 0 in the equation and solve for x. Substitute y = 0 in the equation: o\(x^{2} + 4(0)^{2} = 16\) Simplifying, we get: \(x^{2} = 16\) Thus, \(x = \pm 4\). Therefore, the x-intercepts are (4, 0) and (-4, 0).

Find y-intercepts

Set x = 0 in the equation and solve for y. Substitute x = 0 in the equation: o\((0)^{2} + 4y^{2} = 16\) Simplifying, we get: \(4y^{2} = 16\) Thus, \(y^{2} = 4\) Therefore, \(y = \pm 2\). Thus, the y-intercepts are (0, 2) and (0, -2).

Key concepts

x-intercepts

In a coordinate plane, the x-intercept is where a graph crosses the x-axis. To determine the x-intercepts of an equation, we set y to 0 and solve for x. This is because any point on the x-axis has a y-coordinate of 0.

Let's take the equation from our example: \(x^2 + 4y^2 = 16\)

Setting y to 0, we get:

\x^2 + 4(0)^2 = 16\

Simplifying further:
\x^2 = 16\

Solving for x, we take the square root of both sides:
\x = \pm 4\

So, the x-intercepts are at (4, 0) and (-4, 0).

y-intercepts

Similarly, the y-intercept is the point where the graph intersects the y-axis. To find the y-intercepts, we set x to 0 and solve for y, since any point on the y-axis has an x-coordinate of 0.

Again, using our example equation: \(x^2 + 4y^2 = 16\)

Set x to 0:
\0^2 + 4y^2 = 16\

Simplifying:
\4y^2 = 16\

Divide both sides by 4:
\y^2 = 4\

Solving for y, we take the square root of both sides:
\y = \pm 2\

Therefore, the y-intercepts are at (0, 2) and (0, -2).

solving quadratic equations

Solving quadratic equations involves finding the values of x (or y) that satisfy the equation. Quadratics are equations of the form \(ax^2 + bx + c = 0\).

There are several methods to solve quadratics:

• Factoring
• Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]
• Completing the square

In our example equation \(x^2 + 4y^2 = 16\), although it looks different, it simplifies into a solvable form when finding intercepts.

By setting either x or y to 0, we transformed it into a simpler quadratic form: \(x^2 = 16\) and \(4y^2 = 16\), which we solved by isolating the variable and taking square roots.

coordinate geometry

Coordinate geometry, also known as analytic geometry, involves using an algebraic approach to solve geometric problems.

The coordinate plane is divided into four quadrants by the intersection of the x-axis and y-axis. Any point in this plane can be specified by an ordered pair (x, y).

In our example, we used the principles of coordinate geometry to find where the given equation's graph intersects the x and y axes. By understanding and manipulating these coordinates, we can easily determine intercepts and graph equations accurately.