What Is The $y$-intercept Of The Graph Of The Equation $y = X^2 - 5x + 4$?

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Introduction

The $y$-intercept of a graph is the point at which the graph intersects the $y$-axis. In other words, it is the value of $y$ when $x = 0$. To find the $y$-intercept of the graph of the equation $y = x^2 - 5x + 4$, we need to substitute $x = 0$ into the equation and solve for $y$.

Understanding the Equation

The given equation is a quadratic equation in the form $y = ax^2 + bx + c$, where $a = 1$, $b = -5$, and $c = 4$. This type of equation represents a parabola, which is a U-shaped curve that opens upwards or downwards. The graph of the equation $y = x^2 - 5x + 4$ will be a parabola that opens upwards, since the coefficient of $x^2$ is positive.

Finding the $y$-Intercept

To find the $y$-intercept, we need to substitute $x = 0$ into the equation. This means that we need to replace every instance of $x$ with $0$ and simplify the resulting expression.

import sympy as sp

x = sp.symbols('x')



equation = x**2 - 5*x + 4



y_intercept = equation.subs(x, 0)


print(y_intercept)

When we run this code, we get the following output:

4

This means that the $y$-intercept of the graph of the equation $y = x^2 - 5x + 4$ is $y = 4$.

Graphing the Equation

To visualize the graph of the equation, we can use a graphing tool or software. Here is a graph of the equation $y = x^2 - 5x + 4$:

Graph of the equation y = x^2 - 5x + 4

As we can see from the graph, the $y$-intercept is indeed at the point $(0, 4)$.

Conclusion

In this article, we have found the $y$-intercept of the graph of the equation $y = x^2 - 5x + 4$. We have used the concept of quadratic equations and the properties of parabolas to find the $y$-intercept. We have also graphed the equation to visualize the $y$-intercept. The $y$-intercept is an important concept in mathematics, and it has many applications in various fields such as physics, engineering, and economics.

Applications of the $y$-Intercept

The $y$-intercept has many applications in various fields. Here are a few examples:

  • In physics, the $y$-intercept can be used to find the initial velocity of an object.
  • In engineering, the $y$-intercept can be used to design and optimize systems such as bridges and buildings.
  • In economics, the $y$-intercept can be used to model and analyze economic systems.

Final Thoughts

In conclusion, the $y$-intercept is an important concept in mathematics that has many applications in various fields. We have found the $y$-intercept of the graph of the equation $y = x^2 - 5x + 4$ using the concept of quadratic equations and the properties of parabolas. We have also graphed the equation to visualize the $y$-intercept. The $y$-intercept is a fundamental concept in mathematics that has many practical applications.

References

Further Reading

  • [1] Algebra and Trigonometry by Michael Sullivan
  • [2] Calculus by Michael Spivak
  • [3] Linear Algebra and Its Applications by Gilbert Strang

Introduction

In our previous article, we discussed the concept of the $y$-intercept of a graph and how to find it using the equation $y = x^2 - 5x + 4$. In this article, we will answer some frequently asked questions about the $y$-intercept and provide additional information to help you understand this concept better.

Q&A

Q1: What is the $y$-intercept of a graph?

A1: The $y$-intercept of a graph is the point at which the graph intersects the $y$-axis. It is the value of $y$ when $x = 0$.

Q2: How do I find the $y$-intercept of a graph?

A2: To find the $y$-intercept of a graph, you need to substitute $x = 0$ into the equation of the graph and solve for $y$.

Q3: What is the difference between the $y$-intercept and the $x$-intercept?

A3: The $y$-intercept is the point at which the graph intersects the $y$-axis, while the $x$-intercept is the point at which the graph intersects the $x$-axis. In other words, the $y$-intercept is the value of $y$ when $x = 0$, while the $x$-intercept is the value of $x$ when $y = 0$.

Q4: Can the $y$-intercept be negative?

A4: Yes, the $y$-intercept can be negative. For example, if the equation of the graph is $y = -x^2 + 5x - 4$, the $y$-intercept would be $y = -4$.

Q5: How do I graph a function with a $y$-intercept?

A5: To graph a function with a $y$-intercept, you need to find the $y$-intercept by substituting $x = 0$ into the equation of the function. Then, you can use this point as a reference to draw the graph of the function.

Q6: Can the $y$-intercept be a complex number?

A6: Yes, the $y$-intercept can be a complex number. For example, if the equation of the graph is $y = x^2 + 5x + 4$, the $y$-intercept would be $y = 4 + 5i$, where $i$ is the imaginary unit.

Q7: How do I find the $y$-intercept of a graph with a quadratic equation?

A7: To find the $y$-intercept of a graph with a quadratic equation, you need to substitute $x = 0$ into the equation and solve for $y$. This will give you the $y$-intercept of the graph.

Q8: Can the $y$-intercept be a function of another variable?

A8: Yes, the $y$-intercept can be a function of another variable. For example, if the equation of the graph is $y = x^2 + 5x + 4 + 2t$, where $t$ is a parameter, the $y$-intercept would be a function of $t$.

Conclusion

In this article, we have answered some frequently asked questions about the $y$-intercept of a graph and provided additional information to help you understand this concept better. We hope that this article has been helpful in clarifying any doubts you may have had about the $y$-intercept.

Further Reading

References

  • [1] Algebra and Trigonometry by Michael Sullivan
  • [2] Calculus by Michael Spivak
  • [3] Linear Algebra and Its Applications by Gilbert Strang