Stretching Parabolas And More Completing The Square Homework Answers _TOP_
Stretching Parabolas and More Completing the Square Homework Answers
If you are struggling with stretching parabolas and completing the square homework problems, you are not alone. These topics can be challenging for many students, especially if they are not familiar with the concepts and techniques involved. In this article, we will explain what stretching parabolas and completing the square mean, how to apply them to different types of quadratic equations, and how to find the correct answers for your homework assignments.
stretching parabolas and more completing the square homework answers
What are Stretching Parabolas and Completing the Square?
A parabola is a type of curve that has a U-shape and can be represented by a quadratic equation of the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a parabola can be stretched or compressed vertically by changing the value of a. For example, if a is positive and greater than 1, the parabola will be narrower than the standard parabola y = x^2. If a is positive and less than 1, the parabola will be wider than the standard parabola. If a is negative, the parabola will be upside down.
Completing the square is a method of rewriting a quadratic equation in a form that makes it easier to find its roots, vertex, and axis of symmetry. The idea is to add and subtract a term that makes the expression inside the parentheses a perfect square. For example, consider the equation y = x^2 + 6x + 5. To complete the square, we need to find a number that when added to 6x and divided by 2 gives a perfect square. In this case, that number is 9, because (6x + 9)/2 = (3x + 3)^2. So we add and subtract 9 to get y = (x^2 + 6x + 9) - 9 + 5. Then we simplify and factor the expression inside the parentheses to get y = (x + 3)^2 - 4. This form is called the vertex form of a quadratic equation, because it shows us that the vertex of the parabola is at (-3,-4).
How to Stretch Parabolas and Complete the Square for Different Types of Quadratic Equations?
To stretch a parabola, we need to multiply its equation by a constant factor. For example, if we want to stretch the parabola y = x^2 by a factor of 2, we need to multiply both sides by 2 to get y = 2x^2. This will make the parabola narrower and steeper. Similarly, if we want to compress the parabola by a factor of 1/2, we need to multiply both sides by 1/2 to get y = (1/2)x^2. This will make the parabola wider and flatter.
To complete the square for different types of quadratic equations, we need to follow these steps:
If the coefficient of x^2 is not 1, divide both sides by that coefficient.
Add and subtract a term that makes the expression inside the parentheses a perfect square.
Simplify and factor the expression inside the parentheses.
Rewrite the equation in vertex form.
For example, suppose we want to complete the square for the equation y = -4x^2 + 8x - 3. We follow these steps:
Divide both sides by -4 to get y/-4 = x^2 - 2x + 3/4.
Add and subtract 1 to get y/-4 = (x^2 - 2x + 1) - 1 + 3/4.
Simplify and factor to get y/-4 = (x - 1)^2 - 1/4.
Rewrite in vertex form to get y = -4(x - 1)^2 + 1.
How to Find the Roots, Vertex, and Axis of Symmetry of a Parabola?
One of the advantages of completing the square is that it allows us to find the roots, vertex, and axis of symmetry of a parabola easily. The roots are the values of x that make y equal to zero. The vertex is the point where the parabola changes direction and has the minimum or maximum value of y. The axis of symmetry is the vertical line that passes through the vertex and divides the parabola into two congruent halves.
To find the roots of a parabola, we need to set y equal to zero and solve for x. If the equation is in vertex form, we can use the zero product property to get x - h = 0 or x - h = -2k/a, where a, h, and k are the constants in the vertex form y = a(x - h)^2 + k. For example, if we have y = 2(x + 3)^2 - 4, we can find the roots by setting y equal to zero and getting x + 3 = 0 or x + 3 = -2(-4)/2. This gives us x = -3 or x = 1 as the roots.
To find the vertex of a parabola, we can use the formula (h,k), where h and k are the constants in the vertex form. For example, if we have y = -4(x - 1)^2 + 1, we can find the vertex by using (1,1) as the formula. This means that the vertex is at (1,1).
To find the axis of symmetry of a parabola, we can use the formula x = h, where h is the constant in the vertex form. For example, if we have y = (x + 3)^2 - 4, we can find the axis of symmetry by using x = -3 as the formula. This means that the axis of symmetry is at x = -3.
How to Check Your Homework Answers?
To check your homework answers, you can use several methods. One method is to plug your answers back into the original equation and see if they make sense. For example, if you found that the roots of y = x^2 + 6x + 5 are -5 and -1, you can plug them back into the equation and see if they make it equal to zero. You should get y = (-5)^2 + 6(-5) + 5 = 0 and y = (-1)^2 + 6(-1) + 5 = 0, which are true statements.
Another method is to graph your equation and see if it matches your answers. For example, if you found that the vertex of y = -4(x - 1)^2 + 1 is at (1,1), you can graph it and see if it looks like a downward opening parabola with a maximum point at <cod
In this article, we have learned what stretching parabolas and completing the square mean, how to apply them to different types of quadratic equations, and how to find the roots, vertex, and axis of symmetry of a parabola. We have also learned how to check our homework answers using various methods. We hope that this article has helped you understand these topics better and improve your math skills. If you have any questions or feedback, please feel free to leave a comment below. d282676c82