Determining increase/decrease intervals for ax^2+bx+c

  • Thread starter peripatein
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In summary: By completing the square, you can rewrite the equation as y=a(x-h)^2+k, where (h,k) is the vertex of the parabola. From this, you can see that the parabola will be increasing for all x values greater than the x-coordinate of the vertex, which is -b/2a. This is because the vertex is the highest point of the parabola, and any x value greater than the x-coordinate of the vertex will result in a higher y value. Therefore, the interval for which the parabola is increasing is x>-b/2a.
  • #1
peripatein
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Hello,

Homework Statement



I am trying to determine on what intervals the parabola y=ax^2+bx+c increases/decreases, without resorting to differentiation.

Homework Equations





The Attempt at a Solution



For the function to be increasing on a certain interval f(x1)>f(x2) for any x1 and x2 on that interval such that x1>x2. Hence, ax1^2+bx1+c>ax2^2+bx2+c. That yields, (x1+x2)>-b/a. How do I derive the expected x>-b/2a from that? That result could be obtained for x1=x2, hence Δ=0, but why is that and why ought it to be used in order to obtain the correct answer?
 
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  • #2
Let x2=x1+Δ and then consider the limes of Δ->0?
 
  • #3
peripatein said:
Hello,

Homework Statement



I am trying to determine on what intervals the parabola y=ax^2+bx+c increases/decreases, without resorting to differentiation.

Homework Equations





The Attempt at a Solution



For the function to be increasing on a certain interval f(x1)>f(x2) for any x1 and x2 on that interval such that x1>x2. Hence, ax1^2+bx1+c>ax2^2+bx2+c. That yields, (x1+x2)>-b/a. How do I derive the expected x>-b/2a from that? That result could be obtained for x1=x2, hence Δ=0, but why is that and why ought it to be used in order to obtain the correct answer?

How about completing the square?
 

Related to Determining increase/decrease intervals for ax^2+bx+c

1. How can I determine the intervals in which a quadratic function is increasing or decreasing?

The intervals of increase or decrease for a quadratic function can be determined by analyzing the sign of the coefficient of the squared term (a) in the quadratic equation ax^2 + bx + c. If a > 0, the function will be increasing in the interval (-infinity, +infinity). If a < 0, the function will be decreasing in the same interval.

2. What is the significance of the coefficient of the squared term in determining intervals of increase or decrease?

The coefficient of the squared term (a) in a quadratic function determines the concavity of the graph and the direction of the parabola. If a > 0, the parabola opens upwards and the function is increasing. If a < 0, the parabola opens downwards and the function is decreasing.

3. Can I determine the intervals of increase or decrease without graphing the function?

Yes, the intervals of increase or decrease for a quadratic function can be determined algebraically by analyzing the sign of the coefficient of the squared term (a). This can be done by factoring the function or by using the quadratic formula to find the x-intercepts.

4. How can I use the first derivative to determine the intervals of increase or decrease?

The first derivative of a function represents its rate of change at any given point. For a quadratic function, the first derivative is a linear function in the form of mx + b, where m is the slope of the tangent line at any given point. The intervals of increase or decrease can be determined by analyzing the sign of the first derivative. If the first derivative is positive, the function is increasing, and if it is negative, the function is decreasing.

5. Is it possible for a quadratic function to have multiple intervals of increase or decrease?

Yes, it is possible for a quadratic function to have multiple intervals of increase or decrease. This can happen if the function has a local maximum or minimum point, where the function changes from increasing to decreasing or vice versa. These points can be identified by setting the first derivative equal to zero and solving for x.

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