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Chapter 1: Problem 3

Fill in the blanks. The points at which a graph intersects or touches an axis are called the______of the graph.

Short Answer

Expert verified
The term is roots or zeros

Step by step solution

01

Understanding the exercise

The question requires the student to fill in the blank with the correct term from mathematical vocabulary, that represents points at which a line or curve intersects an axis in a graph.
02

Recall the term

The terminology used in the algebra to denote the points where the graph of an equation intersects the x-axis or y-axis is known as the 'roots' or 'zeros' of the graph.
03

Fill in the blanks

Knowing the term, fill in the blank with the term 'roots' or 'zeros'.

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Key Concepts

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

x-intercept
An x-intercept is the point where a graph crosses or touches the x-axis. At this point, the value of the y-coordinate is always zero because the graph is on the x-axis. Understanding x-intercepts helps you find where the entire graph is positioned horizontally.

To find the x-intercept of an equation, you set the y value to zero and solve for x. For example, if you have the function \( y = 2x + 3 \), to find the x-intercept, solve for \( x \) by setting \( y = 0 \):
  • Set \( 2x + 3 = 0 \)
  • Subtract 3 from both sides: \( 2x = -3 \)
  • Divide by 2: \( x = -\frac{3}{2} \)
The x-intercept is \( (-\frac{3}{2}, 0) \). Remember, at this point, the graph meets the x-axis.
y-intercept
The y-intercept is the point where the graph intersects the y-axis. At this location, the value of the x-coordinate is always zero because the graph is on the y-axis. The y-intercept provides crucial information about where the graph starts on the vertical axis.

To find the y-intercept of an equation, set the x value to zero and solve for y. For the same function \( y = 2x + 3 \), find the y-intercept by setting \( x = 0 \):
  • Set \( y = 2(0) + 3 \)
  • Thus, \( y = 3 \)
So, the y-intercept is \( (0, 3) \). This means the graph crosses the y-axis at that point.
roots
Roots are the values of x that make the equation equal to zero. Often, these are the x-values where the graph will intersect the x-axis. In many cases, the terms "roots," "zeros," and "solutions" are used interchangeably, especially in polynomial and quadratic functions.

To find the roots, set the function equal to zero and solve for x, much like finding x-intercepts. If you have the equation \( x^2 - 4x + 4 = 0 \), you'd solve this quadratic equation for x:
  • Factor the equation: \( (x-2)(x-2) = 0 \)
  • Set each factor to zero: \( x-2 = 0 \)
  • Solve for x: \( x = 2 \).
The root (or roots) is \( x = 2 \), showing where the graph touches the x-axis.
zeros
Zeros are another way to describe the roots of a function, which are the input values that result in an output of zero. They determine where the function does not exist above or below the x-axis.

Just like with roots, zeros are found by setting the equation equal to zero and solving for x.

For example, take the function \( f(x) = x^2 - 5x + 6 \), find the zeros by solving \( f(x) = 0 \):
  • Factor the expression: \( (x-2)(x-3) = 0 \)
  • Set each factor to zero: \( x-2 = 0 \) or \( x-3 = 0 \)
  • Solve for x: \( x = 2 \) and \( x = 3 \).
The zeros of the function are \( x = 2 \) and \( x = 3 \), indicating where the graph hits the x-axis.
intersects an axis
When a graph intersects an axis, it means that the graph crosses or touches either the x-axis or the y-axis at specific points. These intersections reveal crucial insights into the behavior of the function represented by the graph.

When a graph intersects the x-axis, it is at a point where the y-value is zero, simplifying to finding the x-intercept or roots.
  • This is important for observing where the function changes direction or balances out, especially for parabolic shapes.
Conversely, when the graph intersects the y-axis, this occurs at a point where the x-value is zero, revealing the y-intercept.
  • This point often represents the starting value of a graph before any x-values alter the function.
Understanding where a graph intersects an axis is fundamental to analyzing and predicting the function's progression visually.

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Most popular questions from this chapter

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