Quadratic Graphs

• A quadratic function is a polynomial function of degree two, generally expressed in the form $y = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants and $a \neq 0$. The condition $a \neq 0$ is essential because if 'a' were zero, the term $ax^2$ would vanish, resulting in a linear function rather than a quadratic one.

• The graph of a quadratic function is a distinctive U-shaped or N-shaped curve called a parabola. This curve is always smooth and possesses a unique vertical axis of symmetry that divides it into two mirror-image halves.

• The sign of the leading coefficient 'a' dictates the parabola's orientation. If $a > 0$, the parabola opens upwards, forming a 'u-shape', and its turning point is a minimum point. Conversely, if $a < 0$, the parabola opens downwards, forming an 'n-shape', and its turning point is a maximum point.

• The turning point of a parabola, whether a minimum or maximum, is known as the vertex. This point is crucial as it represents the extreme value (either lowest or highest) that the quadratic function can attain.

• Key points on a quadratic graph include the y-intercept, where the graph crosses the y-axis, and the x-intercepts, also known as roots, where the graph crosses or touches the x-axis. A quadratic graph will always have one y-intercept but can have zero, one, or two x-intercepts.

• The standard form of a quadratic equation is $y = ax^2 + bx + c$. This form is particularly useful for quickly identifying the y-intercept, which is always at the point $(0, c)$, as setting $x=0$ simplifies the equation to $y=c$.

• The vertex form of a quadratic equation is $y = a(x - p)^2 + q$. This form directly reveals the coordinates of the vertex as $(p, q)$. The value 'p' also indicates the equation of the axis of symmetry, which is $x = p$. Note that 'p' takes the opposite sign from what appears inside the parenthesis.

• The factored form of a quadratic equation is $y = a(x - x_1)(x - x_2)$. This form is highly useful for identifying the x-intercepts (roots) of the parabola, which are located at $(x_1, 0)$ and $(x_2, 0)$. These are the points where the graph crosses the x-axis, as setting $y=0$ directly yields $x=x_1$ or $x=x_2$.

Y-intercept

• The y-intercept is the point where the parabola crosses the y-axis. It can always be found by setting $x = 0$ in any form of the quadratic equation.

• In the standard form $y = ax^2 + bx + c$, the y-intercept is simply $(0, c)$. This makes the standard form convenient for quickly locating this point.

X-intercepts (Roots)

• The x-intercepts, or roots, are the points where the parabola crosses or touches the x-axis. These are found by setting $y = 0$ and solving the quadratic equation $ax^2 + bx + c = 0$.

• Roots can be found by factoring the quadratic expression, using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, or by completing the square. The number of real roots (zero, one, or two) depends on the value of the discriminant $b^2 - 4ac$.

Vertex

• The vertex is the parabola's turning point, representing its minimum or maximum value. If the equation is in vertex form $y = a(x - p)^2 + q$, the vertex coordinates are directly $(p, q)$.

• For a quadratic in standard form $y = ax^2 + bx + c$, the x-coordinate of the vertex can be calculated using the formula $x = -b/(2a)$. Once the x-coordinate is found, substitute this value back into the original equation to determine the corresponding y-coordinate of the vertex.

• The vertical line passing through the vertex, with the equation $x = -b/(2a)$ (or $x=p$ in vertex form), is the axis of symmetry for the parabola.

• To sketch a quadratic graph, begin by drawing and labeling the x and y axes. This provides the fundamental coordinate system for your plot.

• Identify and mark the y-intercept on the y-axis. This is the point $(0, c)$ from the standard form or by substituting $x=0$ into the equation.

• Find all x-intercepts (roots) by setting $y=0$ and solving the quadratic equation. Mark these points on the x-axis. If there are no real roots, the parabola will not cross the x-axis.

• Determine the overall shape of the parabola by observing the sign of the coefficient 'a'. A positive 'a' indicates a u-shape (opening upwards), while a negative 'a' indicates an n-shape (opening downwards).

• Finally, sketch a smooth curve that passes through all identified intercepts. If the vertex coordinates are known, ensure the curve turns smoothly at this point, respecting the axis of symmetry. Avoid drawing sharp corners or straight line segments, as parabolas are inherently smooth.

Using Vertex and Another Point

• If the vertex $(p, q)$ and at least one other point $(x, y)$ on the parabola are known, the most efficient approach is to use the vertex form $y = a(x - p)^2 + q$. First, substitute the coordinates of the vertex $(p, q)$ into this form.

• Next, substitute the coordinates of the additional known point $(x, y)$ into the partially formed equation. This will create an equation with only 'a' as an unknown, which can then be solved to find the value of the leading coefficient 'a'. Once 'a' is determined, write out the complete quadratic equation.

Using Roots and Another Point

• If the x-intercepts (roots) $(x_1, 0)$ and $(x_2, 0)$ are known, along with another point $(x, y)$ on the parabola, the factored form $y = a(x - x_1)(x - x_2)$ is the preferred choice. Begin by substituting the values of $x_1$ and $x_2$ into this form.

• Subsequently, substitute the coordinates of the additional known point $(x, y)$ into the equation. This allows for the calculation of the coefficient 'a'. After finding 'a', the full quadratic equation can be written, and if required, expanded into standard form.

• The number of x-intercepts a quadratic graph has is determined by the discriminant, $b^2 - 4ac$. If $b^2 - 4ac > 0$, there are two distinct real roots. If $b^2 - 4ac = 0$, there is exactly one real root (a repeated root), meaning the parabola touches the x-axis at its vertex. If $b^2 - 4ac < 0$, there are no real roots, and the parabola does not intersect the x-axis.

• All parabolas exhibit vertical symmetry about their axis of symmetry, which is a vertical line passing through the vertex. This property means that for any point $(x_v + d, y)$ on the parabola, there is a corresponding point $(x_v - d, y)$ at the same height, where $x_v$ is the x-coordinate of the vertex and 'd' is a horizontal distance.

• The domain of any quadratic function is all real numbers, meaning 'x' can take any value from negative infinity to positive infinity. The range, however, is restricted by the vertex. For a u-shaped parabola ($a>0$), the range is $y \ge q$ (where $q$ is the y-coordinate of the vertex), and for an n-shaped parabola ($a<0$), the range is $y \le q$.

• Verify the Shape: Always check the sign of the 'a' coefficient immediately to confirm the correct opening direction (u-shape for $a>0$, n-shape for $a<0$). A mismatch between your calculation and the expected shape indicates an error.

• Double-Check Intercepts and Vertex: Carefully calculate the y-intercept, x-intercepts (if any), and the vertex coordinates. These are the most common points for errors, especially with signs or arithmetic. Use the quadratic formula accurately for roots and $x = -b/(2a)$ for the vertex's x-coordinate.

• Smooth Curve, Not Lines: Ensure your sketch is a smooth, continuous curve, not a series of straight line segments. A parabola's curvature is a defining characteristic, and drawing it with sharp angles will lose marks.

• Choose the Right Form: When deriving a quadratic equation from a graph, select the appropriate form (vertex form or factored form) based on the information provided. This minimizes algebraic manipulation and potential errors. For instance, if the vertex is clearly given, the vertex form is usually simpler.

• Label Key Points: Always label the coordinates of the y-intercept, x-intercepts, and the vertex on your sketch. This demonstrates a complete understanding of the graph's features and helps in verifying the accuracy of your plot.