Where Does $f(x) = 3x^2 - 11x - 4$ Intersect The $x$-axis?The Negative \$x$[/tex\]-intercept Is At $\square, 0$.

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Introduction

When a function intersects the x-axis, it means that the function's value is equal to zero at that point. In other words, the function crosses the x-axis at a specific point. To find where the function $f(x) = 3x^2 - 11x - 4$ intersects the x-axis, we need to find the values of x for which the function is equal to zero.

Setting the Function Equal to Zero

To find the x-intercepts of the function, we set the function equal to zero and solve for x. This means that we need to find the values of x that make the function $f(x) = 3x^2 - 11x - 4$ equal to zero.

f(x)=3x2−11x−4=0f(x) = 3x^2 - 11x - 4 = 0

Using the Quadratic Formula

The equation $3x^2 - 11x - 4 = 0$ is a quadratic equation, which can be solved using the quadratic formula. The quadratic formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a = 3, b = -11, and c = -4. Plugging these values into the quadratic formula, we get:

x=−(−11)±(−11)2−4(3)(−4)2(3)x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(3)(-4)}}{2(3)}

x=11±121+486x = \frac{11 \pm \sqrt{121 + 48}}{6}

x=11±1696x = \frac{11 \pm \sqrt{169}}{6}

x=11±136x = \frac{11 \pm 13}{6}

Finding the x-Intercepts

Now that we have the solutions to the quadratic equation, we can find the x-intercepts of the function. The x-intercepts are the values of x that make the function equal to zero.

x=11+136=246=4x = \frac{11 + 13}{6} = \frac{24}{6} = 4

x=11−136=−26=−13x = \frac{11 - 13}{6} = \frac{-2}{6} = -\frac{1}{3}

Conclusion

In conclusion, the function $f(x) = 3x^2 - 11x - 4$ intersects the x-axis at two points: x = 4 and x = -1/3. These are the x-intercepts of the function.

The Negative x-Intercept

The negative x-intercept is at $-\frac{1}{3}, 0$.

The Positive x-Intercept

The positive x-intercept is at $4, 0$.

Graphing the Function

To visualize the function and its x-intercepts, we can graph the function on a coordinate plane. The graph of the function will be a parabola that opens upward, with the x-intercepts at x = 4 and x = -1/3.

Applications of the Function

The function $f(x) = 3x^2 - 11x - 4$ has many applications in mathematics and other fields. For example, it can be used to model the motion of an object under the influence of gravity, or to find the maximum or minimum value of a function.

Real-World Examples

Here are a few real-world examples of how the function $f(x) = 3x^2 - 11x - 4$ can be used:

  • Projectile Motion: The function can be used to model the motion of a projectile under the influence of gravity. For example, if a ball is thrown upwards with an initial velocity of 10 m/s, the function can be used to find the maximum height reached by the ball.
  • Optimization: The function can be used to find the maximum or minimum value of a function. For example, if a company wants to maximize its profits, the function can be used to find the optimal price and quantity to sell.
  • Engineering: The function can be used in engineering to design and optimize systems. For example, if a engineer wants to design a bridge that can withstand a certain amount of weight, the function can be used to find the optimal shape and size of the bridge.

Conclusion

In conclusion, the function $f(x) = 3x^2 - 11x - 4$ intersects the x-axis at two points: x = 4 and x = -1/3. These are the x-intercepts of the function. The function has many applications in mathematics and other fields, and can be used to model real-world phenomena such as projectile motion, optimization, and engineering design.

Introduction

In our previous article, we discussed how to find the x-intercepts of the function $f(x) = 3x^2 - 11x - 4$. In this article, we will answer some frequently asked questions about the function and its x-intercepts.

Q: What is the x-intercept of the function $f(x) = 3x^2 - 11x - 4$?

A: The x-intercepts of the function are x = 4 and x = -1/3.

Q: How do I find the x-intercepts of the function $f(x) = 3x^2 - 11x - 4$?

A: To find the x-intercepts of the function, you need to set the function equal to zero and solve for x. This can be done using the quadratic formula.

Q: What is the quadratic formula?

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

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to find the x-intercepts of the function $f(x) = 3x^2 - 11x - 4$?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. In this case, a = 3, b = -11, and c = -4. Plugging these values into the formula, you get:

x=−(−11)±(−11)2−4(3)(−4)2(3)x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(3)(-4)}}{2(3)}

x=11±121+486x = \frac{11 \pm \sqrt{121 + 48}}{6}

x=11±1696x = \frac{11 \pm \sqrt{169}}{6}

x=11±136x = \frac{11 \pm 13}{6}

Q: What are the x-intercepts of the function $f(x) = 3x^2 - 11x - 4$?

A: The x-intercepts of the function are x = 4 and x = -1/3.

Q: How do I graph the function $f(x) = 3x^2 - 11x - 4$?

A: To graph the function, you need to plot the x-intercepts on a coordinate plane and draw a parabola that opens upward.

Q: What are some real-world applications of the function $f(x) = 3x^2 - 11x - 4$?

A: The function has many real-world applications, including modeling projectile motion, optimization, and engineering design.

Q: How do I use the function $f(x) = 3x^2 - 11x - 4$ to model projectile motion?

A: To use the function to model projectile motion, you need to plug in the values of the initial velocity, angle of projection, and time of flight into the function.

Q: How do I use the function $f(x) = 3x^2 - 11x - 4$ to optimize a system?

A: To use the function to optimize a system, you need to plug in the values of the system's parameters into the function and find the maximum or minimum value of the function.

Q: How do I use the function $f(x) = 3x^2 - 11x - 4$ in engineering design?

A: To use the function in engineering design, you need to plug in the values of the system's parameters into the function and find the optimal shape and size of the system.

Conclusion

In conclusion, the function $f(x) = 3x^2 - 11x - 4$ intersects the x-axis at two points: x = 4 and x = -1/3. These are the x-intercepts of the function. The function has many real-world applications, including modeling projectile motion, optimization, and engineering design.