Write A Quadratic Function From Its Vertex And Another Point.A Parabola Opening Up Or Down Has Vertex { (1, -1)$}$ And Passes Through { \left(4, -\frac{25}{16}\right)$}$. Write Its Equation In Vertex Form.Simplify Any Fractions.
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Introduction
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will discuss how to write a quadratic function from its vertex and another point.
Vertex Form of a Quadratic Function
The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form is useful when we know the vertex of the parabola and want to write its equation.
Given Information
We are given that the vertex of the parabola is (1, -1) and it passes through the point (4, -25/16). We can use this information to write the equation of the parabola in vertex form.
Writing the Equation
To write the equation of the parabola, we need to find the values of a, h, and k. We are given that the vertex is (1, -1), so we know that h = 1 and k = -1. Now, we need to find the value of a.
Using the Given Point
We are given that the parabola passes through the point (4, -25/16). We can substitute this point into the equation f(x) = a(x - h)^2 + k to find the value of a.
f(4) = a(4 - 1)^2 + (-1) -25/16 = a(3)^2 - 1 -25/16 = 9a - 1 -25/16 + 1 = 9a -9/16 = 9a a = -9/16 * 1/9 a = -1/16
Writing the Equation in Vertex Form
Now that we have found the value of a, we can write the equation of the parabola in vertex form.
f(x) = -1/16(x - 1)^2 - 1
Simplifying the Equation
We can simplify the equation by multiplying the numerator and denominator of the fraction by 16.
f(x) = -(x - 1)^2/16 - 1 f(x) = -(x^2 - 2x + 1)/16 - 1 f(x) = -x^2/16 + x/8 - 1/16 - 1 f(x) = -x^2/16 + x/8 - 1/16 - 16/16 f(x) = -x^2/16 + x/8 - 17/16
Conclusion
In this article, we discussed how to write a quadratic function from its vertex and another point. We used the vertex form of a quadratic function and substituted the given point into the equation to find the value of a. We then wrote the equation of the parabola in vertex form and simplified it.
Example Problems
Problem 1
Write the equation of a parabola in vertex form that has a vertex at (2, 3) and passes through the point (5, 2).
Solution
To write the equation of the parabola, we need to find the values of a, h, and k. We are given that the vertex is (2, 3), so we know that h = 2 and k = 3. Now, we need to find the value of a.
f(5) = a(5 - 2)^2 + 3 2 = a(3)^2 + 3 2 = 9a + 3 2 - 3 = 9a -1 = 9a a = -1/9
Now that we have found the value of a, we can write the equation of the parabola in vertex form.
f(x) = -1/9(x - 2)^2 + 3
Problem 2
Write the equation of a parabola in vertex form that has a vertex at (-1, 2) and passes through the point (3, 1).
Solution
To write the equation of the parabola, we need to find the values of a, h, and k. We are given that the vertex is (-1, 2), so we know that h = -1 and k = 2. Now, we need to find the value of a.
f(3) = a(3 - (-1))^2 + 2 1 = a(4)^2 + 2 1 = 16a + 2 1 - 2 = 16a -1 = 16a a = -1/16
Now that we have found the value of a, we can write the equation of the parabola in vertex form.
f(x) = -1/16(x + 1)^2 + 2
Final Answer
The final answer is f(x) = -1/16(x - 1)^2 - 1.
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Introduction
In our previous article, we discussed how to write a quadratic function from its vertex and another point. We used the vertex form of a quadratic function and substituted the given point into the equation to find the value of a. In this article, we will provide a Q&A section to help you better understand the concept.
Q&A
Q1: What is the vertex form of a quadratic function?
A1: The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Q2: How do I find the value of a in the vertex form of a quadratic function?
A2: To find the value of a, you need to substitute the given point into the equation f(x) = a(x - h)^2 + k and solve for a.
Q3: What if the given point is not in the vertex form of the quadratic function?
A3: If the given point is not in the vertex form of the quadratic function, you need to convert it to the vertex form first. You can do this by using the formula f(x) = a(x - h)^2 + k and substituting the given point into the equation.
Q4: Can I use any point to find the value of a in the vertex form of a quadratic function?
A4: No, you cannot use any point to find the value of a in the vertex form of a quadratic function. You need to use a point that is on the parabola.
Q5: How do I simplify the equation of a quadratic function in vertex form?
A5: To simplify the equation of a quadratic function in vertex form, you can multiply the numerator and denominator of the fraction by the denominator.
Q6: What is the final answer to the problem of writing a quadratic function from its vertex and another point?
A6: The final answer is f(x) = -1/16(x - 1)^2 - 1.
Example Problems with Solutions
Problem 1
Write the equation of a parabola in vertex form that has a vertex at (2, 3) and passes through the point (5, 2).
Solution
To write the equation of the parabola, we need to find the values of a, h, and k. We are given that the vertex is (2, 3), so we know that h = 2 and k = 3. Now, we need to find the value of a.
f(5) = a(5 - 2)^2 + 3 2 = a(3)^2 + 3 2 = 9a + 3 2 - 3 = 9a -1 = 9a a = -1/9
Now that we have found the value of a, we can write the equation of the parabola in vertex form.
f(x) = -1/9(x - 2)^2 + 3
Problem 2
Write the equation of a parabola in vertex form that has a vertex at (-1, 2) and passes through the point (3, 1).
Solution
To write the equation of the parabola, we need to find the values of a, h, and k. We are given that the vertex is (-1, 2), so we know that h = -1 and k = 2. Now, we need to find the value of a.
f(3) = a(3 - (-1))^2 + 2 1 = a(4)^2 + 2 1 = 16a + 2 1 - 2 = 16a -1 = 16a a = -1/16
Now that we have found the value of a, we can write the equation of the parabola in vertex form.
f(x) = -1/16(x + 1)^2 + 2
Final Answer
The final answer is f(x) = -1/16(x - 1)^2 - 1.
Conclusion
In this article, we provided a Q&A section to help you better understand the concept of writing a quadratic function from its vertex and another point. We also provided example problems with solutions to help you practice the concept.
Final Tips
- Make sure to read the problem carefully and understand what is being asked.
- Use the vertex form of a quadratic function to write the equation of the parabola.
- Substitute the given point into the equation to find the value of a.
- Simplify the equation of the quadratic function in vertex form.
By following these tips, you should be able to write a quadratic function from its vertex and another point with ease.