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Chapter 2: Problem 3

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ x^{2}+y^{2}=16 $$

Short Answer

Expert verified
The x-intercepts are at (4, 0) and (-4, 0), and the y-intercepts are at (0, 4) and (0, -4).

Step by step solution

01

Identify the equation

Given the equation \(x^{2}+y^{2}=16\)
02

Find the x-intercepts

To find the x-intercepts, set \(y=0\) in the equation and solve for \(x\).\( x^{2} + 0^{2} = 16 $$x^{2} = 16\)Taking the square root of both sides, \(x = \text{卤}4\)
03

Find the y-intercepts

To find the y-intercepts, set \(x=0\) in the equation and solve for \(y\).\(0^{2} + y^{2} = 16$$y^{2} = 16\)Taking the square root of both sides, \(y = \text{卤}4\)
04

Summarize the intercepts

The x-intercepts are at \((4, 0)\) and \((-4, 0)\). The y-intercepts are at \((0, 4)\) and \((0, -4)\).

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Key Concepts

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

Quadratic Equations
Quadratic equations are mathematical expressions where the highest exponent of the variable is squared (\text{to the power of 2}). They are often written in the general form: \[ ax^2 + bx + c = 0 \], where \( a, b, \text{and} c \) are constants.
  • In this particular exercise, our quadratic equation is in the form of a circle: \[x^2 + y^2 = 16 \].
  • This form can help us understand relationships between \(x \) and \( y \) better.
Quadratic equations can sometimes be tricky, but remember that finding intercepts helps break them down into more manageable parts. This happens particularly when dealing with circles or parabolas.
Coordinate Geometry
Coordinate geometry involves graphs and lines on a coordinate plane. A coordinate plane has an \( x \)-axis (horizontal) and a \( y \)-axis (vertical), intersecting at the origin (0,0).
  • Each point on this plane can be identified by a pair of numbers \((x,y) \)
  • In our example, the equation \(x^2 + y^2 = 16\) graphs as a circle with its center at the origin and radius 4.
Understanding this basic setup helps in visualizing how shapes like circles and parabolas behave on a graph.
Intercept Finding
Intercepts are crucial in understanding how graphs behave and interact with axes. To find an intercept, you simply determine where the graph intersects the \( x \)-axis or \( y \)-axis.
  • For the \( x \)-intercepts, set \( y \) to zero and solve for \( x \). In the equation given: \[ x^2 + 0^2 = 16 \rightarrow x^2 = 16 \rightarrow x = \text{卤}4\]. This results in intercepts at (4,0) and (-4,0).
  • For the \( y \)-intercepts, set \( x \) to zero and solve for \( y \). Here: \[ 0^2 + y^2 = 16 \rightarrow y^2 = 16 \rightarrow y = \text{卤}4\]. This results in intercepts at (0,4) and (0,-4).
Finding intercepts simplifies the analysis of any graph, especially quadratic equations where the graph often intersects axes at multiple points.

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Most popular questions from this chapter

a. Given \(m(x)=4 x^{2}+2 x-3,\) find \(m(-x)\). b. Find \(-m(x)\). c. Is \(m(-x)=m(x)\) ? d. Is \(m(-x)=-m(x)\) ? e. Is this function even, odd, or neither?

Given \(h(x)=\frac{1}{x-6},\) evaluate \((h \circ h)(x)\) and write the domain in interval notation.

Determine if the function is even, odd, or neither. $$ z(x)=\sqrt{49+x^{2}} $$

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(-0.5) $$

Refer to the functions \(f\) and \(g\) and evaluate the functions for the given values of \(x\). \(f=\\{(2,4),(6,-1),(4,-2),(0,3),(-1,6)\\} \quad\) and \(\quad g=\\{(4,3),(0,6),(5,7),(6,0)\\}\) $$(g \circ f)(0)$$
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