A Quadratic Function \[$ F \$\] Is Given:$\[ F(x) = -x^2 + 16x \\](a) Express \[$ F \$\] In Standard Form:$\[ F(x) = -(x-8)^2 + 64 \\](b) Find The Vertex And \[$ X \$\]- And \[$ Y \$\]-intercepts Of

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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 a specific quadratic function, f(x)=−x2+16x{ f(x) = -x^2 + 16x }, and explore its standard form, vertex, and intercepts.

Expressing the Quadratic Function in Standard Form

To express the quadratic function f(x)=−x2+16x{ f(x) = -x^2 + 16x } in standard form, we need to complete the square. The standard form of a quadratic function is f(x)=a(x−h)2+k{ f(x) = a(x - h)^2 + k }, where (h,k){ (h, k) } is the vertex of the parabola.

Step 1: Factor out the coefficient of the squared term

The coefficient of the squared term is -1, so we can factor it out:

f(x)=−(x2−16x){ f(x) = -(x^2 - 16x) }

Step 2: Add and subtract the square of half the coefficient of the linear term

The coefficient of the linear term is 16, so half of it is 8. The square of 8 is 64. We add and subtract 64 inside the parentheses:

f(x)=−(x2−16x+64−64){ f(x) = -(x^2 - 16x + 64 - 64) }

Step 3: Rewrite the expression as a squared term minus 64

Now we can rewrite the expression as a squared term minus 64:

f(x)=−(x2−16x+64)+64{ f(x) = -(x^2 - 16x + 64) + 64 }

Step 4: Factor the squared term

The squared term can be factored as a perfect square:

f(x)=−(x−8)2+64{ f(x) = -(x - 8)^2 + 64 }

This is the standard form of the quadratic function.

Finding the Vertex of the Parabola

The vertex of the parabola is the point (h,k){ (h, k) } in the standard form of the quadratic function. In this case, the vertex is (8,64){ (8, 64) }.

Finding the x-Intercepts of the Parabola

The x-intercepts of the parabola are the points where the graph of the function crosses the x-axis. To find the x-intercepts, we set the function equal to zero and solve for x:

−x2+16x=0{ -x^2 + 16x = 0 }

We can factor out an x:

−x(x−16)=0{ -x(x - 16) = 0 }

This gives us two possible solutions:

x=0{ x = 0 }

or

x−16=0{ x - 16 = 0 }

x=16{ x = 16 }

So, the x-intercepts of the parabola are (0,0){ (0, 0) } and (16,0){ (16, 0) }.

Finding the y-Intercept of the Parabola

The y-intercept of the parabola is the point where the graph of the function crosses the y-axis. To find the y-intercept, we set x equal to zero and solve for y:

f(0)=−(0)2+16(0){ f(0) = -(0)^2 + 16(0) }

f(0)=0{ f(0) = 0 }

So, the y-intercept of the parabola is (0,0){ (0, 0) }.

Conclusion

In this article, we expressed a quadratic function in standard form and found its vertex and intercepts. We used the method of completing the square to express the function in standard form and found the vertex to be (8,64){ (8, 64) }. We also found the x-intercepts to be (0,0){ (0, 0) } and (16,0){ (16, 0) } and the y-intercept to be (0,0){ (0, 0) }. This demonstrates the importance of understanding quadratic functions and their properties in mathematics.

References

Further Reading

Introduction

In our previous article, we explored a quadratic function, f(x)=−x2+16x{ f(x) = -x^2 + 16x }, and expressed it in standard form, f(x)=−(x−8)2+64{ f(x) = -(x - 8)^2 + 64 }. We also found the vertex and intercepts of the parabola. 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 can be expressed in the form f(x)=ax2+bx+c{ f(x) = ax^2 + bx + c }, where a{ a }, b{ b }, and c{ c } are constants.

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

A: The standard form of a quadratic function is f(x)=a(x−h)2+k{ f(x) = a(x - h)^2 + k }, where (h,k){ (h, k) } is the vertex of the parabola.

Q: How do I express a quadratic function in standard form?

A: To express a quadratic function in standard form, you need to complete the square. This involves factoring out the coefficient of the squared term, adding and subtracting the square of half the coefficient of the linear term, and rewriting the expression as a squared term minus a constant.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point (h,k){ (h, k) } in the standard form of the quadratic function. It is the maximum or minimum point of the parabola.

Q: How do I find the x-intercepts of a parabola?

A: To find the x-intercepts of a parabola, you need to set the function equal to zero and solve for x. This will give you the points where the graph of the function crosses the x-axis.

Q: How do I find the y-intercept of a parabola?

A: To find the y-intercept of a parabola, you need to set x equal to zero and solve for y. This will give you the point where the graph of the function crosses the y-axis.

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 crosses the x-axis at two or more points.

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 graph of the function opens downward.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the x-intercepts and the vertex of the parabola. You can then use a ruler or a graphing tool to draw the parabola.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We covered topics such as the standard form of a quadratic function, the vertex and intercepts of a parabola, and how to graph a quadratic function. We hope this article has been helpful in understanding quadratic functions and their properties.

References

Further Reading