Quadratic Equations with 2 plots

In summary, to find the values of 'a' and 'q' in the quadratic equation of y = a(x)^2 + q when given two plots, you can substitute the given points for x and y and set them equal to the equation. This will result in two equations with two unknowns, which can then be solved using elimination. This method can also be extended to determine the values of unknowns in other types of equations, such as lines and cubics.
  • #1
Larrytsai
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0
[SOLVED] Quadratic Equations

How would I find what 'a' and 'q' is in the equation of y=a(x)^2+q when I'm given 2 plots?

Example:Find 'a' and 'q' so that a parabola y=ax^2+q passes through each pair.
a)(-3,11)and(4,18)

if you could give me step by step explanation that would be much appreciated.
 
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  • #2
What does it mean for the parabola y = ax2 + q to pass through a point (m, n)? It means that: n = am2 + q, because we substitute m and n for x and y. Do that for the two points you are given (-3, 11) and (4, 18). What equations do you get?
 
  • #3
well if i did substitute then i would get 11=a(-3)^2+q
11=9a+q
18=16a+q

hmmm now i have a feeling i need to cancel out variable by elimination right?

well yeah... i end up with 2 varibales 'a' and 'q'
 
Last edited:
  • #4
Yes, that is exactly what you would do. You have two equations, and two unknowns, which is exactly what you need.

This is an interesting thing to notice. This quadratic equation is in a special form y = ax2 + c. However, if it was in the general form y = ax2 + bx + c, you will find that, after substituting points for x and y, you will have equations in three unknowns, namely a, b, and c! That means you will need three total equations, and three points. In general, three points completely determine a parabola.

You can extend this further. Since a line is always y = ax + b, in two unknowns a and b, you only need two points to define a line. Since a cubic is y = ax3 + bx2 + cx + d is in 4 unknowns (a, b, c, d) you will need four equations, or four points to define a cubic. And so on.

In your question, the quadratic was missing a bx term (or you can think of it as you already knowing that b = 0), so you only have two unknowns (a and c), and you only needed two points.
 
  • #5
oook i got it now thnx alot
 

1. What is a quadratic equation?

A quadratic equation is a mathematical equation that contains one or more terms in which the variable is raised to the second power. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

2. How do you solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2-4ac)) / 2a. Alternatively, you can also factor the equation or use the completing the square method.

3. What are the two types of solutions for a quadratic equation?

The two types of solutions for a quadratic equation are real and complex. Real solutions are values of the variable that make the equation true, and they can be found using the quadratic formula or by factoring. Complex solutions involve imaginary numbers and can only be found using the quadratic formula.

4. How many solutions can a quadratic equation have?

A quadratic equation can have either two, one, or zero solutions. This depends on the value of the discriminant, which is b^2-4ac. If the discriminant is positive, the equation will have two real solutions. If it is zero, the equation will have one real solution. And if it is negative, the equation will have no real solutions.

5. How are quadratic equations used in real life?

Quadratic equations are used in many real-life situations, such as calculating the maximum profit of a business, predicting the trajectory of a projectile, or finding the optimal shape of a bridge or arch. They are also used in various fields of science, including physics, engineering, and economics.

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