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Chapter 0: Problem 58

Find the \(x\) - and \(y\)-intercepts of the graph of the equation. \(y=x^4-25\)

Short Answer

Expert verified
The x-intercepts are \(\sqrt[4]{25}\) and \(-\sqrt[4]{25}\). The y-intercept is -25.

Step by step solution

01

Find the x-intercepts

To find the x-intercepts, set \(y=0\), then solve for \(x\). The equation becomes: \(0=x^4-25\). This can be solved by adding 25 to both sides, then taking the fourth root of each side. This gives: \(x^4=25\), hence \(x=\sqrt[4]{25}\) or \(x=-\sqrt[4]{25}\). Therefore, the x-intercepts are \(\sqrt[4]{25}\) and \(-\sqrt[4]{25}\).
02

Find the y-intercept

To find the y-intercept, set \(x=0\) in the original equation, and solve for \(y\). This gives: \(y=0^4-25=-25\). Therefore, the y-intercept is -25.
03

Verify the intercepts

Plotting the function \(y=x^4-25\) confirms that it intersects the x-axis at the points \(\sqrt[4]{25}\) and \(-\sqrt[4]{25}\), and the y-axis at the point -25. Hence, the previous calculations for the x- and y-intercepts are correct.

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Key Concepts

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

x-intercepts
X-intercepts are the points where a graph crosses the x-axis. These are crucial because they tell us where the function equals zero. To find the x-intercepts of an equation, set the output value, or y, to zero and solve for the variable x.
In our example, the equation was given as:
  • \(y = x^4 - 25\)
By setting \(y=0\), the equation becomes:
  • \(0 = x^4 - 25\)
Solve this equation:
  • Add 25 to both sides: \(x^4 = 25\)
  • Take the fourth root of each side: \(x = \sqrt[4]{25}\) or \(x = -\sqrt[4]{25}\)
These roots represent the x-intercepts of the graph. You can visualize these as the points where the graph touches or crosses the x-axis.
y-intercepts
Y-intercepts are the points where the graph touches or crosses the y-axis. They help us understand the behavior of the function at specific inputs. Finding the y-intercept is straightforward; set the input value, or x, to zero and solve for y.
Using the provided equation:
  • \(y = x^4 - 25\)
  • Let \(x = 0\)
The equation then is:
  • \(y = 0^4 - 25\)
Therefore, \(y = -25\).
The y-intercept of the graph is -25. This is the point on the y-axis where x equals zero, and is crucial for understanding the vertical behavior of the graph.
graphing equations
Graphing equations is a key skill in understanding the behavior and characteristics of functions. It involves plotting points that satisfy the equation and drawing a curve or line through those points. The primary goal of graphing is to see how the function behaves visually.
For the equation \(y = x^4 - 25\), both x- and y-intercepts have been identified:
  • X-intercepts: \(x = \sqrt[4]{25}\) and \(x = -\sqrt[4]{25}\)
  • Y-intercept: \(y = -25\)
These intercepts are helpful starting points in graphing the function accurately.
By plotting these points and noting the general shape dictated by the \(x^4\) term (a polynomial function), you can further understand how the graph behaves above and below the x-axis.
polynomial functions
A polynomial function is a mathematical expression involving a sum of powers of x, each multiplied by a coefficient. Polynomials are smooth, continuous curves defined by their degree, which is the highest power of x in the expression.
The given equation, \(y = x^4 - 25\), is a polynomial as it includes a power of x, specifically, it is of degree 4. This degree gives us some insight into the graph's shape:
  • Because it is even, the ends of the graph will mirror each other.
  • The leading term, \(x^4\), tells us that the graph will be u-shaped open upwards.
Understanding polynomials is essential in algebra and calculus because they make up the foundation of many more complex functions and models.

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Most popular questions from this chapter

Writing In your own words, explain the meanings of domain and range.

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