Determine The Properties Of The Function In The Form $f(x) = Ax^2 + \text{(other Terms)}$ Given The Following Conditions:- The Vertex Is A Maximum.- The $y$-intercept Is Negative.- The $x$-intercepts Are Negative.- The Axis

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will focus on determining the properties of a quadratic function in the form $f(x) = ax^2 + \text{(other terms)}$ given certain conditions.

Conditions for the Quadratic Function

The conditions given for the quadratic function are:

• The vertex is a maximum.
• The $y$-intercept is negative.
• The $x$-intercepts are negative.
• The axis of symmetry is to the left of the $y$-axis.

Understanding the Conditions

Let's break down each condition and understand its implications on the quadratic function.

The Vertex is a Maximum

A quadratic function has a maximum vertex when the coefficient of the $x^2$ term is negative. This means that the parabola opens downwards, and the vertex represents the highest point on the graph.

The $y$-Intercept is Negative

The $y$-intercept of a quadratic function is the point where the graph intersects the $y$-axis. A negative $y$-intercept indicates that the graph lies below the $x$-axis at this point.

The $x$-Intercepts are Negative

The $x$-intercepts of a quadratic function are the points where the graph intersects the $x$-axis. Negative $x$-intercepts indicate that the graph lies to the left of the $y$-axis at these points.

The Axis of Symmetry is to the Left of the $y$-Axis

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. When the axis of symmetry is to the left of the $y$-axis, it means that the vertex lies to the left of the $y$-axis.

Determining the Properties of the Quadratic Function

Given the conditions, we can determine the properties of the quadratic function as follows:

• Coefficient of the $x^2$ term: Since the vertex is a maximum, the coefficient of the $x^2$ term is negative. Let's denote this coefficient as $a$. Therefore, $a < 0$.
• $y$-Intercept: Since the $y$-intercept is negative, we can write the equation of the quadratic function as $f(x) = ax^2 + bx + c$, where $c < 0$.
• $x$-Intercepts: Since the $x$-intercepts are negative, we can write the equation of the quadratic function as $f(x) = ax^2 + bx + c$, where the roots of the equation are negative. This means that the quadratic function can be factored as $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are negative roots.
• Axis of Symmetry: Since the axis of symmetry is to the left of the $y$-axis, we can write the equation of the quadratic function as $f(x) = ax^2 + bx + c$, where the axis of symmetry is given by $x = -\frac{b}{2a}$.

Solving for the Coefficient $a$

We can solve for the coefficient $a$ by using the fact that the vertex is a maximum. Since the vertex is a maximum, the axis of symmetry is given by $x = -\frac{b}{2a}$. Substituting this into the equation of the quadratic function, we get:

$$f(x) = ax^2 + bx + c$$

$$f(-\frac{b}{2a}) = a(-\frac{b}{2a})^2 + b(-\frac{b}{2a}) + c$$

$$f(-\frac{b}{2a}) = \frac{b^2}{4a} - \frac{b^2}{2a} + c$$

Since the vertex is a maximum, we know that $f(-\frac{b}{2a}) > 0$. Therefore, we can write:

$$\frac{b^2}{4a} - \frac{b^2}{2a} + c > 0$$

Simplifying this inequality, we get:

$$-\frac{b^2}{4a} + c > 0$$

Since $a < 0$, we know that $-\frac{b^2}{4a} < 0$. Therefore, we can write:

$$c > 0$$

This means that the $y$-intercept is positive, which contradicts the given condition that the $y$-intercept is negative. Therefore, we must have:

$$a = -1$$

Solving for the Coefficient $b$

We can solve for the coefficient $b$ by using the fact that the $x$-intercepts are negative. Since the $x$-intercepts are negative, we can write the equation of the quadratic function as:

$$f(x) = a(x - r_1)(x - r_2)$$

where $r_1$ and $r_2$ are negative roots. Expanding this equation, we get:

$$f(x) = ax^2 - (ar_1 + ar_2)x + ar_1ar_2$$

Since the $y$-intercept is negative, we know that $ar_1ar_2 < 0$. Therefore, we can write:

$$ar_1ar_2 = -c$$

where $c > 0$. Substituting this into the equation of the quadratic function, we get:

$$f(x) = ax^2 - (ar_1 + ar_2)x - c$$

Since the axis of symmetry is to the left of the $y$-axis, we know that $-\frac{b}{2a} < 0$. Therefore, we can write:

$$-\frac{b}{2a} = -\frac{ar_1 + ar_2}{2a}$$

Simplifying this equation, we get:

$$ar_1 + ar_2 = 0$$

Since $a < 0$, we know that $ar_1 < 0$ and $ar_2 < 0$. Therefore, we can write:

$$r_1 = -r_2$$

Substituting this into the equation of the quadratic function, we get:

$$f(x) = ax^2 - 2ar_1x - c$$

Since the $y$-intercept is negative, we know that $-c < 0$. Therefore, we can write:

$$c > 0$$

This means that the $y$-intercept is positive, which contradicts the given condition that the $y$-intercept is negative. Therefore, we must have:

$$b = -2r_1$$

Solving for the Coefficient $c$

We can solve for the coefficient $c$ by using the fact that the $y$-intercept is negative. Since the $y$-intercept is negative, we can write:

$$c < 0$$

This means that the $y$-intercept is negative, which is consistent with the given condition.

Conclusion

In this article, we determined the properties of a quadratic function in the form $f(x) = ax^2 + \text{(other terms)}$ given certain conditions. We found that the coefficient of the $x^2$ term is negative, the $y$-intercept is negative, the $x$-intercepts are negative, and the axis of symmetry is to the left of the $y$-axis. We also solved for the coefficients $a$, $b$, and $c$ using the given conditions.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Quadratic Equations" by Purplemath
• [3] "Quadratic Formula" by Math Is Fun

Further Reading

• "Quadratic Functions and Equations" by Khan Academy
• "Quadratic Formula and Functions" by IXL
• "Quadratic Equations and Functions" by Mathway
  Quadratic Function Q&A ==========================

Introduction

In our previous article, we discussed the properties of a quadratic function in the form $f(x) = ax^2 + \text{(other terms)}$ given certain conditions. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. It can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What are the properties of a quadratic function?

A: The properties of a quadratic function include:

• The coefficient of the $x^2$ term is negative.
• The $y$-intercept is negative.
• The $x$-intercepts are negative.
• The axis of symmetry is to the left of the $y$-axis.

Q: How do I determine the properties of a quadratic function?

A: To determine the properties of a quadratic function, you can use the following steps:

1. Write the equation of the quadratic function in the form $f(x) = ax^2 + bx + c$.
2. Determine the coefficient of the $x^2$ term, which is $a$.
3. Determine the $y$-intercept, which is $c$.
4. Determine the $x$-intercepts, which are the roots of the equation.
5. Determine the axis of symmetry, which is given by $x = -\frac{b}{2a}$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Write the equation in the form $ax^2 + bx + c = 0$.
2. Determine the coefficient of the $x^2$ term, which is $a$.
3. Determine the coefficient of the $x$ term, which is $b$.
4. Determine the constant term, which is $c$.
5. Use the quadratic formula to solve for the roots of the equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you can follow these steps:

1. Write the equation in the form $ax^2 + bx + c = 0$.
2. Determine the coefficient of the $x^2$ term, which is $a$.
3. Determine the coefficient of the $x$ term, which is $b$.
4. Determine the constant term, which is $c$.
5. Plug the values of $a$, $b$, and $c$ into the quadratic formula.
6. Simplify the expression to find the roots of the equation.

Q: What are the applications of quadratic functions?

A: Quadratic functions have many applications in mathematics, science, and engineering. Some examples include:

• Modeling the motion of objects under the influence of gravity.
• Finding the maximum or minimum value of a function.
• Solving systems of linear equations.
• Finding the roots of a polynomial equation.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We discussed the properties of a quadratic function, how to determine its properties, and how to solve a quadratic equation using the quadratic formula. We also discussed the applications of quadratic functions in mathematics, science, and engineering.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Quadratic Equations" by Purplemath
• [3] "Quadratic Formula" by Math Is Fun

Further Reading

• "Quadratic Functions and Equations" by Khan Academy
• "Quadratic Formula and Functions" by IXL
• "Quadratic Equations and Functions" by Mathway