Quadratic Graphs: Roots and Turning Points

A parabola is perfectly mirrored across a vertical axis of symmetry. This vertical line always passes exactly through the turning point. Because of this symmetry, the $x$-coordinate of the turning point is located exactly halfway between the two roots.

Find the coordinates of the turning point for $y = x^2 - 8x + 12$ using symmetry, then describe its sketch.

Step 1: Find the roots by setting $y = 0$ and factorising.

• $x^2 - 8x + 12 = 0$
• $(x - 2)(x - 6) = 0$

Step 2: Find the midpoint of the roots to locate the axis of symmetry.

• Midpoint $x = \frac{2 + 6}{2} = 4$
• The equation of the line of symmetry is $x = 4$.

Step 3: Substitute the $x$-value back into the original equation to find the $y$-coordinate.

• $y = (4)^2 - 8(4) + 12$
• $y = 16 - 32 + 12 = -4$

Step 4: Sketch the approximate diagram. A proper mathematical sketch is a freehand, smooth curve that does not need exact plotting, but must show key features clearly. Your sketch for this graph must show:

• A U-shaped parabola.
• A $y$-intercept clearly labelled at $(0, 12)$ on the vertical axis (where $x = 0$).
• Two $x$-intercepts clearly labelled at $(2, 0)$ and on the horizontal axis (where ).

Solving Quadratic Equations Algebraically

To find the exact roots of a quadratic equation algebraically, you must first rearrange it into the standard form $ax^2 + bx + c = 0$. From here, you can factorise or use a formula.

Factorisation relies on the null factor law, which states that if two factors multiply to make zero, at least one of them must be zero. Some expressions use the Difference of Two Squares pattern, such as $x^2 - 16 = 0$, which factorises directly into $(x - 4)(x + 4) = 0$. For harder, non-monic equations (), you must split the middle term.

Solve $3x^2 + 10x + 8 = 0$ algebraically by factorisation.

Step 1: Multiply $a$ and $c$ to find the target product.

• $3 \times 8 = 24$.
• The factors of $24$ that add to $10$ are $6$ and $4$.

Step 2: Split the middle term and factorise in pairs.

• $3x^2 + 6x + 4x + 8 = 0$
• $3x(x + 2) + 4(x + 2) = 0$

Step 3: Group the terms into two brackets.

• $(3x + 4)(x + 2) = 0$

Step 4: Apply the null factor law to find the roots.

• $3x + 4 = 0 \Rightarrow x = -\frac{4}{3}$
• $$

When an equation cannot be factorised, you must use the quadratic formula. The expression inside the square root ($b^2 - 4ac$) is called the discriminant and determines how many real roots exist: two if positive, one if zero, and none if negative.

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Calculate the roots of $2x^2 + 5x - 4 = 0$ to 2 decimal places.

Step 1: Identify $a$, $b$, and $c$.

• $a = 2$, $b = 5$, $c = -4$

Step 2: Substitute the values into the formula using brackets for negative numbers.

• $x = \frac{-(5) \pm \sqrt{(5)^2 - 4(2)(-4)}}{2(2)}$

Step 3: Simplify the discriminant and the denominator.

• $x = \frac{-5 \pm \sqrt{25 + 32}}{4}$

Step 4: Calculate the two possible values.

• $x = \frac{-5 + \sqrt{57}}{4} \approx 0.64$

Completing the Square and Turning Points

Writing a quadratic equation in vertex form ($y = a(x + p)^2 + q$) is known as completing the square. This algebraic method instantly reveals the exact coordinates of the turning point.

When reading the coordinates from the completed square bracket $(x + p)^2$, the $x$-coordinate is the negative of the value inside the bracket ($-p$), while the $y$-coordinate matches the value outside exactly (). The turning point is therefore explicitly .

Write $y = x^2 + 6x + 2$ in the form $y = (x + p)^2 + q$, state the turning point, and describe the sketch.

Step 1: Halve the $x$ coefficient to form the bracket.

• Half of $6$ is $3$.
• Bracket form: $(x + 3)^2$

Step 2: Expand the bracket to see what constant it produces.

• $(x + 3)^2 = x^2 + 6x + 9$

Step 3: Adjust the constant to match the original equation.

• We have $+9$ but we need $+2$, so we must subtract $7$.
• Vertex form: $y = (x + 3)^2 - 7$

Step 4: Extract the turning point coordinates.

• Change the sign of $+3$ and keep $-7$.
• The turning point is $(-3, -7)$.

Step 5: Sketch the approximate diagram. Your approximate diagram derived from this method must clearly show:

• A U-shaped parabola.
• A $y$-intercept mathematically identified at $(0, 2)$.
• A minimum vertex explicitly identified and labelled at $(-3, -7)$.
• Two $x$-intercepts. These are found by setting $(x$, giving precise roots labelled at and .

If the discriminant is negative and there are no real roots, the turning point will sit entirely above the $x$-axis (for a U-shape) or below it (for an n-shape). In this case, your sketch must clearly show the curve never crossing the $x$-axis.