Complete Parts (a) And (b) For $f(x) = -1 + 5x - 3x^2$.(a) State The Degree And Leading Coefficient Of $f$. (b) State The End Behavior Of The Graph Of $f$.(a) The Degree Of $f$ Is $\square$ And Its Leading

In this article, we will delve into the properties of a quadratic function, specifically the degree and leading coefficient, as well as the end behavior of its graph. We will use the function $f(x) = -1 + 5x - 3x^2$ as a case study to illustrate these concepts.

Degree and Leading Coefficient of a Quadratic Function

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. In the function $f(x) = -1 + 5x - 3x^2$, the highest power of $x$ is two, so the degree of $f$ is two.

The leading coefficient of a quadratic function is the coefficient of the term with the highest power of the variable. In this case, the leading coefficient is -3, which is the coefficient of the term $-3x^2$.

Calculating the Degree and Leading Coefficient

To calculate the degree and leading coefficient of a quadratic function, we need to look at the terms of the function and identify the term with the highest power of the variable. In the function $f(x) = -1 + 5x - 3x^2$, the term with the highest power of $x$ is $-3x^2$, which has a degree of two. The coefficient of this term is -3, which is the leading coefficient.

End Behavior of a Quadratic Function

The end behavior of a quadratic function refers to the behavior of the function as $x$ approaches positive or negative infinity. In the case of the function $f(x) = -1 + 5x - 3x^2$, we can determine the end behavior by looking at the leading term, which is $-3x^2$.

Since the leading term is negative, the function will approach negative infinity as $x$ approaches positive infinity. Conversely, the function will approach positive infinity as $x$ approaches negative infinity.

Determining the End Behavior

To determine the end behavior of a quadratic function, we need to look at the leading term and determine whether it is positive or negative. If the leading term is positive, the function will approach positive infinity as $x$ approaches positive infinity and negative infinity as $x$ approaches negative infinity. Conversely, if the leading term is negative, the function will approach negative infinity as $x$ approaches positive infinity and positive infinity as $x$ approaches negative infinity.

Graphing a Quadratic Function

The graph of a quadratic function is a parabola that opens upward or downward. The vertex of the parabola is the minimum or maximum point of the function, depending on whether the leading coefficient is positive or negative.

In the case of the function $f(x) = -1 + 5x - 3x^2$, the leading coefficient is negative, so the graph of the function will open downward. The vertex of the parabola will be the maximum point of the function.

Finding the Vertex

To find the vertex of a quadratic function, we need to use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the leading and linear terms, respectively. In the case of the function $f(x) = -1 + 5x - 3x^2$, the leading coefficient is -3 and the linear coefficient is 5.

Plugging these values into the formula, we get $x = -\frac{5}{2(-3)} = \frac{5}{6}$. To find the y-coordinate of the vertex, we need to plug this value of $x$ into the function.

Conclusion

In conclusion, the degree and leading coefficient of a quadratic function can be determined by looking at the terms of the function and identifying the term with the highest power of the variable. The end behavior of a quadratic function can be determined by looking at the leading term and determining whether it is positive or negative. The graph of a quadratic function is a parabola that opens upward or downward, and the vertex of the parabola is the minimum or maximum point of the function.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath

Discussion

What are some other properties of quadratic functions that we can explore? How can we use the degree and leading coefficient to determine the end behavior of a quadratic function? What are some real-world applications of quadratic functions?