Bringing Equations to Life: Sketching Quadratic Graphs

The Anatomy of a Quadratic Graph

A quadratic graph, or parabola, has several key features:

• Turning point (Vertex): The highest or lowest point of the parabola, depending on if the $a$ coefficient is $a > 0$ or $a < 0$.
• Axis of Symmetry: A vertical line that runs through the turning point, dividing the parabola into two mirror-image halves.
• Direction: Determined by the coefficient $a$. If $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards.
• Y-intercept: The point where the parabola crosses the y-axis ($c$).
• Roots or X-intercepts: Points where the parabola intersects the x-axis. Can be real (touching the x-axis) or complex (not touching the x-axis).

 

 

Steps to Sketching Quadratic Graphs

1. Determine the Direction: Look at the sign of $a$ in $y = ax^2 + bx + c$ to know whether the graph opens up or down.
2. Find the Vertex: Use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and substitute it into the equation to find the y-coordinate.
3. Plot the Y-intercept: Identify $c$ in the equation and mark it on the y-axis.
4. Identify the Roots: Solve $ax^2 + bx + c = 0$ to find the roots, if they exist, and plot them on the x-axis.
5. Sketch the Parabola: Using the points found, draw the parabola, making sure it's symmetrical about the axis of symmetry.