Problem in graph of Quadratic Equation

[Neha Sanghvi]

Hey, could someone please help me with this problem? I just want to know why the graph of Quadratic Equation are symmetric and why does the maxima and minima ( = -b/2a ) the mean of the two roots? One more thing I didn't get was what does the a represent in ax2+bx+c ( ax square+bx+c)? I know for b and c, but couldn't figure out for a? Is it for the slope or whether the graph will be > 0 or < 0?

 

[Galileo]

If you write $ax^2+bx+c$ in the more 'insightful' form:

$$a\left(x+\frac{b}{2a}\right)^2+\left(c-\frac{b^2}{4a}\right)$$

You can see that it is equal to the graph of $ax^2$, but shifted vertically by $c-\frac{b^2}{4a}$ and horizontally by $\frac{b}{2a}$. The a can be seen as a scaling factor in the y-direction. If it's positive, you have a 'dale-parabola', if it's negative it's a 'hill-parabola'.

 

[quicknote]

Sure, I would be happy to help you with your questions about the graph of a quadratic equation. First, let's discuss why the graph of a quadratic equation is symmetric. This is because a quadratic equation has an axis of symmetry, which is a vertical line that divides the graph into two equal halves. This axis of symmetry is located at x = -b/2a, which is the x-coordinate of the vertex (or maxima/minima) of the graph. Since the quadratic equation is symmetric about this line, the maxima and minima are equidistant from the axis of symmetry, making them the mean of the two roots.

Next, let's talk about the coefficient a in the general form of a quadratic equation, ax^2+bx+c. The value of a determines the shape and direction of the graph. If a is positive, the graph will open upward, and if a is negative, the graph will open downward. This is because the coefficient a affects the steepness of the curve. A larger value of a will result in a steeper curve, while a smaller value will result in a flatter curve. Therefore, a does not represent the slope of the graph, but rather the direction and shape of the curve.

I hope this helps to clarify your questions about the graph of a quadratic equation. If you have any further questions, please don't hesitate to ask. As a scientist, it is important to understand the fundamental concepts and principles behind mathematical equations and graphs, so I am happy to assist you in any way I can.