Quadratic Graphs

• A quadratic graph is a curve called a parabola, which represents the function $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.

• The leading coefficient ($a$) determines the 'concavity' or direction of the curve: if $a > 0$, the parabola is upright (U-shaped), and if $a < 0$, it is inverted (n-shaped).

• Every parabola possesses a turning point (or vertex), which is the absolute minimum point for an upright parabola or the absolute maximum point for an inverted one.

• The graph is symmetrical about a vertical line passing through the turning point, known as the axis of symmetry.

• The y-intercept is found by evaluating the function at $x = 0$, which always yields the constant term $c$ in the form $y = ax^2 + bx + c$.

• The x-intercepts (or roots) are the values of $x$ for which $y = 0$; these are found by solving the quadratic equation $ax^2 + bx + c = 0$ via factorisation, completing the square, or the quadratic formula.

• The discriminant, $\Delta = b^2 - 4ac$, dictates the number of x-intercepts: if $\Delta > 0$, there are two; if $\Delta = 0$, the turning point touches the axis (one root); if $\Delta < 0$, the graph never crosses the x-axis.

Finding the Turning Point

• Completing the Square: By rewriting the equation in the form $y = a(x + p)^2 + q$, the turning point is immediately identified as $(-p, q)$.
• Symmetry Method: Since the parabola is symmetrical, the x-coordinate of the turning point is exactly halfway between the roots, or calculated as $x = -\frac{b}{2a}$.

Sketching the Graph

• Step 1: Identify the shape (U or n) based on the sign of $a$.
• Step 2: Calculate the y-intercept by setting $x = 0$.
• Step 3: Find the x-intercepts by setting $y = 0$ and solving for $x$.
• Step 4: Determine the turning point coordinates using completing the square or the symmetry formula.

Note: If the quadratic does not factorise easily, use the discriminant first to check if x-intercepts even exist before attempting to solve.

• Roots vs. Intercepts: While 'roots' refers to the solutions of the equation $f(x)=0$, 'x-intercepts' refers to the specific coordinates $(x, 0)$ on the Cartesian plane where the graph meets the horizontal axis.

• Check the Form: Always ensure the equation is in the standard form $y = ax^2 + bx + c$ before identifying $a$, $b$, and $c$. Rearrange if necessary.

• Labeling: In sketching questions, examiners look for four key labels: the y-intercept, any x-intercepts, the turning point, and the general correct shape.

• Verification: Use the discriminant as a quick 'sanity check'. If your algebra produces two roots but the discriminant is negative, you have made a calculation error.

• Calculator Use: Modern calculators can find roots and turning points; use these to verify your manual working, but always show the algebraic steps (like completing the square) to secure method marks.

• Sign Errors in Vertex Form: A common mistake is identifying the turning point of $y = (x - 3)^2 + 5$ as $(-3, 5)$ instead of $(3, 5)$. Remember the formula is $a(x + p)^2 + q$ where the x-coord is $-p$.

• Confusing Intercepts: Students often mix up the x and y intercepts. Always remember: $y$-intercept is where $x=0$, and $x$-intercept is where $y=0$.

• Incomplete Sketches: Drawing a 'V' shape instead of a smooth curve 'U' shape can lose marks for mathematical communication.

• Quadratic Inequalities: Solving $ax^2 + bx + c > 0$ relies heavily on the graph; the solution is the set of x-values where the curve is above the x-axis.

• Calculus Link: The turning point of a quadratic can also be found using differentiation by setting the first derivative $\frac{dy}{dx} = 2ax + b$ to zero.

• Transformations: The vertex form $y = a(x+p)^2 + q$ is a direct result of translating the basic curve $y = x^2$ by vector $\begin{pmatrix} -p \\ q \end{pmatrix}$ and scaling it by $a$.