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Chapter 11: Problem 3

Find the intercepts of the equation \(y^{2}=16-4 x^{2}\).

Short Answer

Expert verified
(0, 4), (0, -4), (2, 0), (-2, 0)

Step by step solution

01

- Finding the y-intercepts

To find the y-intercepts, set the value of x to 0 in the equation. Substitute x=0 in the equation y^2 = 16 - 4(0)^2Simplify the equation to find the value of y.
02

- Solve for y

Simplify the equation from Step 1:y^2 = 16Take the square root of both sides to find y:y = 卤鈭�16Thus, the values of y are 4 and -4.
03

- Finding the x-intercepts

To find the x-intercepts, set the value of y to 0 in the equation. Substitute y=0 in the equation 0 = 16 - 4x^2Simplify the equation to solve for x.
04

- Solve for x

Simplify the equation from Step 3:4x^2 = 16Divide both sides by 4:x^2 = 4Take the square root of both sides to find x:x = 卤鈭�4Thus, the values of x are 2 and -2.
05

- List the intercepts

Combine the results from Steps 2 and 4. The intercepts are at points (0, 4), (0, -4) for y-intercepts and (2, 0), (-2, 0) for x-intercepts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

y-intercepts
To find the y-intercepts of the equation, we need to understand that these are points where the graph crosses the y-axis. At these points, the x-coordinate is always zero.

To find the y-intercepts for the given equation, let's set x to 0 and solve for y:

Given equation: \[ y^2 = 16 - 4x^2 \]Substitute x=0: \[ y^2 = 16 - 4(0)^2 \]Simplify the right-hand side: \[ y^2 = 16 \]Take the square root of both sides: \[ y = 卤鈭�16 \]Calculate the square root: \[ y = 4 \] and \[ y = -4 \]Therefore, the y-intercepts are at the points (0, 4) and (0, -4). These points are where the graph intersects the y-axis.
x-intercepts
Finding the x-intercepts involves determining where the graph crosses the x-axis. At these points, the y-coordinate is always zero.

To find the x-intercepts for the given equation, set y to 0 and solve for x:

Given equation: \[ y^2 = 16 - 4x^2 \]Substitute y=0: \[ 0 = 16 - 4x^2 \]Rearrange the equation: \[ 4x^2 = 16 \]Divide both sides by 4: \[ x^2 = 4 \]Take the square root of both sides: \[ x = 卤鈭�4 \]Calculate the square root: \[ x = 2 \] and \[ x = -2 \]Hence, the x-intercepts are at the points (2, 0) and (-2, 0). These points represent where the graph intersects the x-axis.
quadratic equations
Quadratic equations are polynomial equations of degree 2, typically written in the form \[ ax^2 + bx + c = 0 \]In our exercise, the given equation can also be written in this form by rearranging terms.

A quadratic equation can have two, one, or no real solutions. This is determined by the discriminant, \( b^2 - 4ac \).
  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is one real root (also known as a repeated root).
  • If the discriminant is negative, there are no real roots.
In our example, by manipulating the given equation \[ y^2 = 16 - 4x^2 \], it can also be treated as a quadratic in terms of either x or y.Recall:
  • To find the y-intercepts, set x to 0.
  • To find the x-intercepts, set y to 0.
Knowing these standard approaches helps in solving and understanding the nature of the curve represented by the quadratic equation.

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Most popular questions from this chapter

Find an equation for the hyperbola described. Graph the equation. Foci at (0,-2) and (0,2)\(;\) asymptote the line \(y=-x\)

Solve: \((x+3)^{2}=20\)

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