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10.09 Quadratic functions and technology


Graphing quadratics using technology

We can use a graphing calculator or online tool such as GeoGebra or Desmos to determine the key features of a quadratic graph.

While using the calculator, we still need to be familiar with what the key features are so we know which commands to give our calculator. The exact commands will vary from tool to tool, but the key features will remain the same.

Graphing a quadratic function with the TI-Nspire

Worked example

Question 1

By first determining the $y$y-intercept, $x$x-intercepts and turning point of $y=-0.5(x+1)^2+5$y=0.5(x+1)2+5, sketch a graph of the parabola.

This video will take you through the process of finding each of the key features.

In summary, we have:

  • $y$y-intercept is $(0,4.5)$(0,4.5)
  • $x$x-intercepts are $(-4.16,0)$(4.16,0) and $(2.16,0)$(2.16,0)
  • maximum turning point is $(-1,5)$(1,5)

We can plot these points and join them with a smooth curve.


Solving quadratic equations using technology

Up to this point we have looked at some different ways to solve quadratic equations. A range of these methods are algebraic, meaning we focus on manipulation of the algebraic equation to find the solutions.

If the quadratic equation is simple enough we may be able to find the solution by graphing the function of the quadratic. However, even with simple quadratics it can be difficult to be consistent and neat enough when graphing by hand to read off the vertex and intercepts of a parabola.

Luckily, there are many forms of technology available today that can help us to solve equations both algebraically and graphically. The great thing about using computers when exploring mathematics is that, once we understand and are confident with the concepts, we can let them do all the heavy lifting!


Practice questions

Question 2

Use your calculator or other handheld technology to graph $y=4x^2-64x+263$y=4x264x+263.

Then answer the following questions.

  1. What is the vertex of the graph?

    Give your answer in coordinate form.

    The vertex is $\left(\editable{},\editable{}\right)$(,)

  2. What is the $y$y-intercept?

    Give your answer in coordinate form.

    The $y$y-intercept is $\left(\editable{},\editable{}\right)$(,)

Question 3

An object launched from the ground has a height (in meters) after $t$t seconds that is modeled by the equation $y=-4.9t^2+58.8t$y=4.9t2+58.8t.

Graph this equation using a calculator or other technology then answer the following questions.

  1. What is the maximum height of the object?

  2. After how many seconds is the object at its maximum height?

  3. After how many seconds does the object return to the ground?

Question 4

We want to solve the equation $2x\left(x-\frac{5}{2}\right)=3$2x(x52)=3.

  1. Rewrite the left side of the equation as a function.

  2. Rewrite the right side of the equation as a function.

  3. Graph both functions using the graphing functionality of your graphics calculator. Hence, solve the equation $2x\left(x-\frac{5}{2}\right)=3$2x(x52)=3 for $x$x.

    Write all solutions together on the same line, separated by commas.

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