The graph of a quadratic function is called a parabola.
The parabolais a U-shaped curvethat can open upwardsor downwards.
Graphing Quadratic Functions
The graph of a quadratic function \$f(x) = ax^2 + bx + c\$ can be sketched by:
- Identifying the vertex.
- Finding the intercepts.
- Plotting additional points.
- Drawing the parabola.
Vertex
The vertex of a parabola is its highest or lowest point.
The vertex of a quadratic function \$f(x) = ax^2 + bx + c\$ is the point:
\$\$(h, k) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\$\$
The vertexis the turning pointof the parabola.
Intercepts
\$x\$-Intercepts
The \$x\$-intercepts are the points where the parabola crosses the \$x\$-axis.
The \$x\$-intercepts of a quadratic function \$f(x) = ax^2 + bx + c\$ are the solutions to the equation \$ax^2 + bx + c = 0\$.
For a quadratic function\$f(x) = ax^2 + bx + c\$, the \$y\$-interceptis \$c\$.
Additional Points
To sketch the parabola, it is helpful to plot additional points on either side of the vertex.
Choose symmetricalpointsaround the vertexto ensurethe parabolais accurate.
Drawing the Parabola
Once the vertex, intercepts, and additional points are plotted, draw a smooth curve through the points to complete the parabola.
1. If \$a > 0\$, the parabolaopensupwards. 2. If \$a < 0\$, the parabolaopensdownwards. 3. The largerthe absolutevalueof \$a\$, the narrowerthe parabola. 4. The smallerthe absolutevalueof \$a\$, the widerthe parabola.
1. The vertexis \$(h, k)\$. 2. The coefficient\$a\$ determinesthe shapeand directionof the parabola.
1. Graph the quadratic function \$f(x) = -3x^2 + 6x - 2\$. 2. Identify the vertex, \$x\$-intercepts, and \$y\$-intercept. 3. How does the coefficient \$a\$ affect the shape and direction of the parabola?
How do different representations of quadratic functions (standard form, vertex form) influence our understanding of their graphs?
