Solving Quadratic Problem: y=a(x^2)+bx+c

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In summary, the conversation discussed solving for the coefficients in a quadratic equation in the form of y=a(x^2)+bx+c using a known root, y-intercept, and point. The conversation also touched on setting up a system of equations to solve for the coefficients.
  • #1
Cmunro
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I have been given a root of 4, a y-intercept of 12 and a known point of (2,8). I have then been asked to write the equation in the form of y=a(x^2)+bx+c

I am assuming the relevant equations are: y=a(x-"alpha")(x-"beta")

Ok, so I know that c=12, but I can't see how I draw the a and b part of the general form from the facts that I have been given. I tried the y=a(x-"alpha")(x-"beta") equation in hopes that I could then expand it out to general form. In this equation alpha=4, however when I substitute y and x with the point (2,8) I am still left with 2 variables: beta and a.

What am I missing?

Thanks, Cat
 
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  • #2
You have done everything correctly, the next step is to set up a system of equations using the data you have been given. You know that when you substitute the root in for x, the result must be zero. When you substitute 2 in for x, the result must be 8 [from your point (2,8)];

[tex]\left.\begin{array}{rcr}
16a + 4b + 12 & = & 0\\
4b + 2b + 12 & = & 8
\end{array}\right\}
[/tex]

From this you should be able to solve for the coefficients of x (a & b). Can you go from here?
 
Last edited:
  • #3
I can go on from here. Thank you very much!
 
  • #4
Cmunro said:
I can go on from here. Thank you very much!
My pleasure :smile:
 

1. What is a quadratic function?

A quadratic function is a mathematical function that can be written in the form y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. It is a type of polynomial function that contains a squared term.

2. How do you graph a quadratic function?

To graph a quadratic function, you can plot points on a coordinate plane or use the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the x-intercepts. You can also find the y-intercept by plugging in x = 0 and solving for y. Once you have a few points, you can connect them to create a parabola.

3. What are the key features of a quadratic function?

The key features of a quadratic function are the vertex, which is the highest or lowest point on the parabola depending on whether the function opens upward or downward, the x-intercepts, which are the points where the parabola crosses the x-axis, and the y-intercept, which is the point where the parabola crosses the y-axis.

4. How do you solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a or factor the equation if possible. If the equation cannot be factored, you can use the quadratic formula to find the solutions, or roots, of the equation. The solutions will be the x-values where the parabola crosses the x-axis.

5. How can you use a quadratic function to model real-life situations?

Quadratic functions can be used to model a variety of real-life situations, such as the height of a ball thrown into the air, the profit of a business, or the trajectory of a rocket. By finding the vertex and x-intercepts of the function, you can determine the maximum or minimum value of the situation and the points where certain outcomes occur. This can be useful in making predictions and analyzing data in various fields of science and engineering.

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