Algebra I

Graphing Quadratic Equations

Big Picture

Quadratic functions involve a variable being squared. The graphs of quadratic functions are parabolas that are usually transformed versions of the basic equation \(y = x^2\). Graphing quadratic functions give us more information about the x-intercepts, which are also the solutions to quadratic equations. We can also know the number of solutions to a quadratic equation by graphing it.

Key Terms

Quadratic Function: A nonlinear function that can be written as \(y = ax^2 + bx + c\), where \(a ≠ 0\).

Parabola: The U-shaped graph formed by a quadratic function.

Vertex: The lowest or highest point on the parabola.

Axis of Symmetry: The vertical line that divides the parabola into two symmetric parts (mirror images).

x-Intercept: The point where a line crosses the x-axis.

Quadratic Equation: A nonlinear equation that can be written as \(ax^2 + bx + c = 0\) where \(a ≠ 0\).

Root (of a polynomial): The value that makes the polynomial equal to \(0\).

Zero (of a function): The value that makes the function equal to \(0\).

Quadratic Functions

The standard form of a quadratic function: \(y = ax^2 + bx + c, a ≠ 0\).

When graphed, all quadratic functions are parabolas. The values of \(a\), \(b\), and \(c\) affect how the parabola looks.

  • The value of a stretches or shrinks the graph
  • \(a > 1\) makes the graph skinnier
  • \(a < 1\) makes the graph fatter
  • If a is negative, the graph curves downward.
  • If a is positive, the vertex is at the bottom of the parabola and is the minimum value of the function.
  • If a is negative, the vertex is at the top of the parabola and is the maximum value of the function.
  • \(c\) is a constant that shifts the parabola up or down.
  • Increasing \(c\) moves the parabola up; decreasing \(c\) moves the parabola down.
  • The point \((0, c)\) is on the parabola.
  • \(b\) is a constant that shifts the parabola left and right. The axis of symmetry is the line \(x = -\frac{b}{2a}\).
  • Increasing \(b\) moves the parabola to the left; decreasing b moves the parabola to the right.

Algebra I

Graphing Quadratic Equations cont.

Graphing Quadratic Functions

The most straightforward way to graph quadratic functions is to make a table of values, plot them, then connect the points with a smooth curve.

Intercept Form

Another way to graph quadratic functions is to rewrite the function in the intercept form.

  • Intercept form: \(y = a(x-p)(x-q)\), where \(p\) and \(q\) are the x-intercepts.

The intercept form is just the factored form of the quadratic function.

  • Using the zero-product property, when \(y = 0, x = p \text{ or } x = q\).
  • \((p, 0)\) and \((q, 0)\) are the x-intercepts.

The intercept form is useful because it tells us:

  • The x-intercepts
  • The axis of symmetry is halfway between the two intercepts, so the axis of symmetry is \(x = \frac{p + q}{2}\)
  • Knowing the axis of symmetry, we can find the vertex by plugging \(x = \frac{p + q}{2}\) into the function
  • If \(a > 0\), the graph opens up; if \(a < 0\), the graph opens down

With these points, we can draw the parabola.

Knowing the x-intercepts and one other point on the parabola, we can also write the quadratic function for the parabola.

Solving Quadratic Equations by Graphing

Finding the roots or zeroes of a quadratic equation is the same as solving the quadratic equation; we want to find the values of \(x\) that make the equation equal to \(0\).

  • This is the same as finding the x-intercepts.

A quadratic equation is the same as a quadratic function where \(y = 0\).

The x-intercepts can be found by factoring the equation (rewritten in the intercept form) or by graphing.

Looking at the graph tells us the number of solutions to the quadratic equation.

  • The equation has \(2\) solutions if the graph has \(2\) x-intercepts (see graph on left).
  • The equation has \(1\) solution if the graph has only \(1\) x-intercept (see graph in middle).
  • The equation has no solutions if the graph has no x-intercepts (see graph on right).

Notes