Given The Quadratic Function $y = X^2 + 8x + C$, Where $c$ Is A Real Number Constant, Answer The Following Questions.(a) If $c = 4$, Find The $x$-intercepts Of The Function In Simplest Radical Form. [3 Points](b) Use

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Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will focus on solving quadratic equations of the form y=x2+8x+cy = x^2 + 8x + c, where cc is a real number constant.

Understanding the Quadratic Function

The quadratic function y=x2+8x+cy = x^2 + 8x + c is a parabola that opens upwards or downwards, depending on the value of cc. The graph of the function is a U-shaped curve that has a single vertex. The x-intercepts of the function are the points where the graph intersects the x-axis.

Finding x-Intercepts of the Function

To find the x-intercepts of the function, we need to set y=0y = 0 and solve for xx. This is because the x-intercepts are the points where the graph intersects the x-axis, and at these points, the value of yy is zero.

(a) If c=4c = 4, find the xx-intercepts of the function in simplest radical form.

To find the x-intercepts of the function when c=4c = 4, we need to substitute c=4c = 4 into the equation and solve for xx. The equation becomes:

y=x2+8x+4y = x^2 + 8x + 4

We set y=0y = 0 and solve for xx:

x2+8x+4=0x^2 + 8x + 4 = 0

To solve this quadratic equation, we can use the quadratic formula:

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

In this case, a=1a = 1, b=8b = 8, and c=4c = 4. Substituting these values into the quadratic formula, we get:

x=−8±82−4(1)(4)2(1)x = \frac{-8 \pm \sqrt{8^2 - 4(1)(4)}}{2(1)}

Simplifying the expression under the square root, we get:

x=−8±64−162x = \frac{-8 \pm \sqrt{64 - 16}}{2}

x=−8±482x = \frac{-8 \pm \sqrt{48}}{2}

x=−8±432x = \frac{-8 \pm 4\sqrt{3}}{2}

x=−4±23x = -4 \pm 2\sqrt{3}

Therefore, the x-intercepts of the function when c=4c = 4 are x=−4+23x = -4 + 2\sqrt{3} and x=−4−23x = -4 - 2\sqrt{3}.

Conclusion

In this article, we have discussed the quadratic function y=x2+8x+cy = x^2 + 8x + c and found the x-intercepts of the function when c=4c = 4. We used the quadratic formula to solve the quadratic equation and found the x-intercepts in simplest radical form. The x-intercepts of the function are x=−4+23x = -4 + 2\sqrt{3} and x=−4−23x = -4 - 2\sqrt{3}.

References

Further Reading

Introduction

In our previous article, we discussed the quadratic function y=x2+8x+cy = x^2 + 8x + c and found the x-intercepts of the function when c=4c = 4. In this article, we will answer some frequently asked questions about quadratic functions.

Q&A

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It is a function of the form y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are real numbers.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is the minimum or maximum point of the graph, depending on the value of aa.

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=−b2ax = -\frac{b}{2a}. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can substitute the x-coordinate into the equation of the function.

Q: What is the x-intercept of a quadratic function?

A: The x-intercept of a quadratic function is the point on the graph where the function intersects the x-axis. It is the point where the value of yy is zero.

Q: How do I find the x-intercept of a quadratic function?

A: To find the x-intercept of a quadratic function, you can set y=0y = 0 and solve for xx. This will give you the x-coordinate of the x-intercept.

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

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. A quadratic function has a parabolic shape, while a linear function has a straight line shape.

Q: Can a quadratic function have more than one x-intercept?

A: Yes, a quadratic function can have more than one x-intercept. This occurs when the graph of the function intersects the x-axis at more than one point.

Q: How do I determine the number of x-intercepts of a quadratic function?

A: To determine the number of x-intercepts of a quadratic function, you can use the discriminant, which is given by the formula b2−4acb^2 - 4ac. If the discriminant is positive, the function has two x-intercepts. If the discriminant is zero, the function has one x-intercept. If the discriminant is negative, the function has no x-intercepts.

Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. This occurs when the value of aa is negative.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the points on the graph.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We have discussed the vertex, x-intercept, and discriminant of a quadratic function, and we have provided examples of how to find these values.

References

Further Reading