Graph a quadratic equation

Key Notes :

Understanding Quadratic Equations and Graphs

  • A quadratic equation is a polynomial equation of degree two, typically expressed in the form ax² + bx + c = 0.
  • The graph of a quadratic equation is a parabola, which is a U-shaped curve.

Key Properties of Quadratic Graphs

  • Vertex: The point where the parabola changes direction.
  • Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves.
  • Roots: The points where the parabola intersects the x-axis (also known as x-intercepts).
  • Y-intercept: The point where the parabola intersects the y-axis.

Steps to Graph a Quadratic Equation

  1. Identify the Vertex: Use the vertex formula: (-b/2a, f(-b/2a)) to find the coordinates of the vertex.
  2. Determine the Axis of Symmetry: The axis of symmetry is a vertical line passing through the vertex. Its equation is x = -b/2a.
  3. Find the Roots (if any): Solve the quadratic equation to find the x-intercepts. You can use factoring, completing the square, or the quadratic formula.
  4. Plot Key Points: Plot the vertex, the axis of symmetry, and any roots you found.
  5. Sketch the Parabola: Use the shape of the parabola (upward if a > 0, downward if a < 0) and the plotted points to sketch the curve.

Example

Graph the quadratic equation f(x) = x² – 4x + 3

  • Vertex: (-b/2a, f(-b/2a)) = (2, -1)
  • Axis of Symmetry: x = 2
  • Roots: Solve x² – 4x + 3 = 0 to find x = 1 and x = 3
  • Sketch the parabola using the vertex, axis of symmetry, and roots.

Additional Tips

  • To get a more accurate graph, plot additional points by substituting different x-values into the equation and calculating the corresponding y-values.
  • Consider the end behavior of the parabola: if a > 0, the parabola opens upward, and if a < 0, the parabola opens downward.
  • Use graphing technology (e.g., graphing calculators or online graphing tools) to visualize the graph more easily

Learn with an example

Graph the function f(x)=6x2.

Plot the vertex. Then plot another point on the parabola. If you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

To graph the parabola, first find the vertex.

f(x)=6x2

=6(x–0)2+0

The vertex is (0,0), which is the origin.

Now look for another point on the parabola with integer or half-integer coordinates. One such point is (1/2,3/2). (Plugging in x=1/2 yields y=3/2.) So, plot this point.

Graph the function f(x)=7x2.

Plot the vertex. Then plot another point on the parabola. If you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

To graph the parabola, first find the vertex.

f(x)=7x2

=7(x–0)2+0

The vertex is (0,0), which is the origin.

Now look for another point on the parabola with integer or half-integer coordinates. One such point is (1,7). (Plugging in x=1 yields y=7.) So, plot this point.

Graph the function f(x)=5x2.

Plot the vertex. Then plot another point on the parabola. If you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

To graph the parabola, first find the vertex.

f(x)=5x2

=5(x–0)2+0

The vertex is (0,0), which is the origin.

Now look for another point on the parabola with integer or half-integer coordinates. One such point is (1,5). (Plugging in x=1 yields y=5.) So, plot this point.

let’s practice!