Graphs quadratic graphs and gradient of the graphh

Here are some notes on quadratic graphs and the gradient of a graph:

Quadratic Graphs
A quadratic graph is the graphical representation of a quadratic function, which is a polynomial of degree 2. Its general form is:

$$y = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants and $a \neq 0$.
The graph of a quadratic function is a parabola.

• If $a > 0$, the parabola opens upwards (U-shaped) and has a minimum point.
• If $a < 0$, the parabola opens downwards (inverted U-shaped) and has a maximum point. Key features include:
• The vertex (the minimum or maximum point). Its $x$-coordinate is given by $x = -\frac{b}{2a}$.
• The axis of symmetry, which is a vertical line passing through the vertex, $x = -\frac{b}{2a}$.
• The y-intercept, found by setting $x=0$, which is $(0, c)$.
• The x-intercepts (or roots), found by setting $y=0$ and solving $ax^2 + bx + c = 0$. These are the points where the parabola crosses the $x$-axis.

Gradient of a Graph (for a curve)
For a straight line, the gradient is constant. However, for a curve, the gradient changes from point to point.

• The gradient of a curve at a specific point is defined as the gradient of the tangent line to the curve at that point.
• To find the gradient of a curve $y = f(x)$ at any point, we use differentiation. The derivative of the function, denoted as $\frac{dy}{dx}$ or $f'(x)$, gives the gradient function.
• For a function of the form $y = ax^n$, the derivative is:

$$\frac{dy}{dx} = nax^{n-1}$$

• To find the gradient at a specific point $(x_1, y_1)$, substitute $x_1$ into the gradient function $\frac{dy}{dx}$.
• A positive gradient means the curve is increasing at that point.
• A negative gradient means the curve is decreasing at that point.
• A zero gradient means the curve has a horizontal tangent at that point, indicating a stationary point (maximum, minimum, or point of inflection).