Example 3: $f(x) = 12x - 9 - 3x^2$Standard Form: $\qquad$a =$ $\qquad$ $b =$ $\qquad$ $c =$ $\qquad$Does It Open Up Or Down? $\qquad$Vertex: $\qquad$Axis Of

Introduction

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will delve into the world of quadratic functions, focusing on the standard form, vertex, and axis of symmetry. We will use the example function $f(x) = 12x - 9 - 3x^2$ to illustrate the concepts and provide a deeper understanding of quadratic functions.

Standard Form

The standard form of a quadratic function is given by:

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

where $a$, $b$, and $c$ are constants. In the example function $f(x) = 12x - 9 - 3x^2$, we can identify the values of $a$, $b$, and $c$ as follows:

• $a = -3$: The coefficient of the $x^2$ term, which determines the direction and width of the parabola.
• $b = 12$: The coefficient of the $x$ term, which determines the horizontal shift of the parabola.
• $c = -9$: The constant term, which determines the vertical shift of the parabola.

Does it Open Up or Down?

The value of $a$ determines whether the parabola opens up or down. If $a > 0$, the parabola opens up, and if $a < 0$, the parabola opens down. In this case, $a = -3$, which means the parabola opens down.

Vertex

The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola, depending on whether it opens up or down. To find the vertex, we can use the formula:

$$x = -\frac{b}{2a}$$

Plugging in the values of $a$ and $b$, we get:

$$x = -\frac{12}{2(-3)} = 2$$

To find the $y$-coordinate of the vertex, we can plug this value of $x$ into the original function:

$$f(2) = 12(2) - 9 - 3(2)^2 = 15 - 12 = 3$$

Therefore, the vertex of the parabola is at the point $(2, 3)$.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is the line of symmetry for the parabola, and it divides the parabola into two equal parts. The equation of the axis of symmetry is given by:

$$x = -\frac{b}{2a}$$

In this case, the equation of the axis of symmetry is:

$$x = 2$$

Graphing the Parabola

To graph the parabola, we can use the vertex and the axis of symmetry as reference points. We can start by plotting the vertex at the point $(2, 3)$. Then, we can use the axis of symmetry to plot two points on either side of the vertex. For example, we can plot the point $(0, f(0))$ and the point $(4, f(4))$. By connecting these points with a smooth curve, we can obtain the graph of the parabola.

Conclusion

In conclusion, quadratic functions are an essential concept in mathematics, and they have numerous applications in various fields. The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The value of $a$ determines whether the parabola opens up or down, and the vertex and axis of symmetry are crucial in graphing the parabola. By understanding these concepts, we can gain a deeper insight into the world of quadratic functions and their applications.

Example Problems

1. Find the vertex and axis of symmetry of the parabola given by $f(x) = 2x^2 + 5x - 3$.
2. Graph the parabola given by $f(x) = -x^2 + 4x + 2$.
3. Find the equation of the axis of symmetry of the parabola given by $f(x) = x^2 - 2x - 3$.

Solutions

1. To find the vertex and axis of symmetry, we can use the formulas:

   $$x = -\frac{b}{2a}$$

   $$y = f(x)$$

   Plugging in the values of $a$ and $b$, we get:

   $$x = -\frac{5}{2(2)} = -\frac{5}{4}$$

   $$y = f(-\frac{5}{4}) = 2(-\frac{5}{4})^2 + 5(-\frac{5}{4}) - 3 = \frac{25}{8} - \frac{25}{4} - 3 = -\frac{19}{8}$$

   Therefore, the vertex of the parabola is at the point $(-\frac{5}{4}, -\frac{19}{8})$, and the equation of the axis of symmetry is:

   $$x = -\frac{5}{4}$$

2. To graph the parabola, we can use the vertex and the axis of symmetry as reference points. We can start by plotting the vertex at the point $(1, 3)$. Then, we can use the axis of symmetry to plot two points on either side of the vertex. For example, we can plot the point $(-1, f(-1))$ and the point $(3, f(3))$. By connecting these points with a smooth curve, we can obtain the graph of the parabola.

3. To find the equation of the axis of symmetry, we can use the formula:

   $$x = -\frac{b}{2a}$$

   Plugging in the values of $a$ and $b$, we get:

   $$x = -\frac{-2}{2(1)} = 1$$

   Therefore, the equation of the axis of symmetry is:

   x = 1$<br/>

Frequently Asked Questions

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means it has the general form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I determine whether a quadratic function opens up or down?

A: To determine whether a quadratic function opens up or down, you need to look at the value of $a$. If $a > 0$, the parabola opens up, and if $a < 0$, the parabola opens down.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula:

$$x = -\frac{b}{2a}$$

Then, plug this value of $x$ into the original function to find the $y$-coordinate of the vertex.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is the line of symmetry for the parabola, and it divides the parabola into two equal parts.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the vertex and the axis of symmetry as reference points. Start by plotting the vertex, then use the axis of symmetry to plot two points on either side of the vertex. By connecting these points with a smooth curve, you can obtain the graph of the parabola.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have numerous applications in various fields, including physics, engineering, and economics. Some common applications include:

• Modeling the motion of objects under the influence of gravity
• Describing the shape of a parabola
• Finding the maximum or minimum value of a function
• Solving optimization problems

Q: How do I solve a quadratic equation?

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

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

This formula will give you two solutions for the equation.

Q: What is the difference between a quadratic function and a quadratic equation?

A: A quadratic function is a polynomial function of degree two, while a quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$. While a quadratic function can be used to model a parabola, a quadratic equation is used to solve for the value of $x$ that makes the equation true.

Q: Can you provide some examples of quadratic functions?

A: Yes, here are a few examples of quadratic functions:

• $f(x) = x^2 + 3x - 4$
• $f(x) = 2x^2 - 5x + 1$
• $f(x) = -x^2 + 2x + 3$

Q: Can you provide some examples of quadratic equations?

A: Yes, here are a few examples of quadratic equations:

• $x^2 + 3x - 4 = 0$
• $2x^2 - 5x + 1 = 0$
• $-x^2 + 2x + 3 = 0$

Q: How do I determine the vertex of a quadratic function from its graph?

A: To determine the vertex of a quadratic function from its graph, you can look for the point where the parabola changes direction. This point is the vertex of the parabola.

Q: How do I determine the axis of symmetry of a quadratic function from its graph?

A: To determine the axis of symmetry of a quadratic function from its graph, you can look for the vertical line that passes through the vertex of the parabola. This line is the axis of symmetry of the parabola.

Q: Can you provide some tips for graphing quadratic functions?

A: Yes, here are a few tips for graphing quadratic functions:

• Start by plotting the vertex of the parabola.
• Use the axis of symmetry to plot two points on either side of the vertex.
• Connect these points with a smooth curve to obtain the graph of the parabola.
• Use a ruler or a straightedge to draw the graph of the parabola.

Q: Can you provide some tips for solving quadratic equations?

A: Yes, here are a few tips for solving quadratic equations:

• Use the quadratic formula to solve the equation.
• Simplify the expression under the square root sign.
• Use the plus-or-minus sign to find two solutions for the equation.
• Check your solutions by plugging them back into the original equation.