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Topic outline

• Graph of a Quadratic Function

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  • • In this subtopic, you will learn about graphing quadratic functions.  The graph of a quadratic is a parabola, which may open upwards or downwards. The vertex or turning point of the parabola is an important feature, as are the $x$-intercepts (if it has any) and the $y$-intercept (there will always be one).
      

      
      

      Our resident engineer, Dr Kim Blackmore, explains why knowing about quadratic functions is important for engineers:

      "Quadratic functions have always been a personal favourite of mine  - they have a nice parabola shape and are symmetrical, which appeals to me!  In engineering, we use quadratic functions to optimise performance. We examine the various inputs and output measurements and then try to determine at what point the maximum efficiency occurs - which will be the turning point of the quadratic function. For example, in chemical engineering, quadratic functions can be used to find the optimal amount of a fuel additive to use to improve the performance of a vehicle."

      
      

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      Watch

      Graphs of Quadratic Equations: part a 22m 43s

      MATH 1076 - Graph of quadratic functions_ part a_default_04a3968c

      
      

      
      

      
      
      

      Graphs of Quadratic Equations: part b 24m 42s

      MATH 1076 - Graph of quadratic functions_ part b_default_c0f4001f

      
      

      
      

      
      
      

      Vertex Form 29m 50s

    • MATH 1076 - Vertex form_default

      
      

      
      
      
      
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       Quadratic Functions

      We have met quadratic expressions several times already in this course. 

      • We have solved quadratic equations of the form $ax^2 + bx + c = 0$. 
      • We have solved quadratic inequalities like $ax^2 + bx + c \geq 0$.

      (Note that $a \neq 0$.)
      

      
      

      We will now concentrate on understanding and graphing quadratic functions that can be expressed as:

      $f : x \rightarrow ax^2 + bx + c$

      or

      $f(x) = ax^2 + bx + c.$

      Quadratic functions can be used to describe situations where the speed or rate of change of an object changes as it moves. The motion of projectiles (rockets, balls, paper planes) are good examples of situations that can be modelled with quadratic functions. To sketch a graph of a quadratic function we find the important features, namely the $x$-intercepts (if any), the $y$-intercept, and the vertex. From these, we can sketch the function, and find the equation in vertex form: $y=a(x-h)^2+k$.
      

      
      
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       The ‘standard form’ of a quadratic

      A quadratic function in one variable, say $x$, can be written in the form $f(x) = ax^2+bx+c$ where $a, b, c$ are real numbers and $a \neq 0$. This is called the standard form. We can label the output $y$, so quadratic functions can also be expressed as $y = ax^2 + bx + c$. We will examine the effect of each of the parameters, $a, b,$ and $c$.
      
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       The effect of $a$. 

      We will start by setting parameters $b$ and $c$ to zero and examining the effect of $a$.
      

      The shape of each curve is a parabola. Each has a turning point or vertex. If the graph opens up, it is called concave up and the vertex is a minimum. If the graph opens down, it is called concave down and the vertex is a maximum.
      

    • Activity 6E - The Effect of $a$

      What
      Use this activity to check your understanding of the effect of $a$ in quadratic equations. 

      How

      • Manipulate the sliders on the graph below to display $y=x^2$, $y=-x^2$, and $y=2x^2$. You should check your results using a table of values.
      • Then,
      1. State the domain and range of each of the three functions
      2. Identify the turning points of the quadratics
      3. Describe the effects of parameter $a$

      
      

    • answer (click to expand)

      	Domain	Range	Turning Point/vertex
      $f(x)=x^2$	$(-\infty, \infty)$	$[0, \infty)$	$(0,0)$
      $f(x)=-x^2$	$(-\infty, \infty)$	$(-\infty, 0]$	$(0,0)$
      $f(x)=2x^2$	$(-\infty, \infty)$	$[0, \infty)$	$(0,0)$

      
      So,
      If $a<0$ vertex is maximum
      If $a>0$ vertex is minimum
      
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       The effect of $c$. 

      Now we will set parameter $a$ to $1$, parameter $b$ to zero, and examine the effect of $c$.
      

      Standard form allows us to easily identify the vertical intercept ($y$-intercept) of the graph of the quadratic function.
      

    • Activity 6F - The Effect of $c$

      What
      Use this activity to check your understanding of the effect of $c$ in quadratic equations. 

      How

      • Manipulate the sliders on the graph below to display $y=x^2$, $y=x^2+3$, and $y=x^2-4$. You should check your results using a table of values.
      • Then,
      1. State the domain and range of each of the three functions
      2. Describe the effects of adding a constant term $c$

      
      

    • answer (click to expand)

      	Domain	Range	Turning Point/vertex
      $y=x^2$	$(-\infty, \infty)$	$[0, \infty)$	$(0,0)$
      $y=x^2+3$	$(-\infty, \infty)$	$[3, \infty)$	$(0,3)$
      $y=x^2-4$	$(-\infty, \infty)$	$[-4, \infty)$	$(0,-4)$

      
      So,
      for positive $c$, vertex moves 3 units upfrom (0,0)
      for negative $c$, vertex moves 3 units down from (0,0)
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       The 'Factored Form' of a Quadratic

      A quadratic is expressed in factored form if it is written as $y = a(x − r)(x − s)$ where $a, r, s$ are constants and $a \neq 0$. The values $r$ and $s$ are called zeros of the quadratic.
      
    • Activity 6G - Factored Form

      What
      Complete the question to check your knowledge of factored form. 

      How

      • Enter the response to the question in the space provided.
    • Factored form allows us to easily identify the horizontal intercepts ($x$-intercepts) of the graph of the quadratic function. Not all quadratics can be written in factored form.
      

    • Activity 6H - Factored Form

      What
      This activity contains questions which can be used to self-check your understanding of factored form.

      How

      • Select the correct response to each of the multiple-choice questions.
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       Graphing

      We can modify the function $y = ax^2$in another way. Suppose we replace $x$ by $x − h$. The function is $y = a(x − h) ^2$. What is the effect?
      
      
      
    • Activity 6I - The Effect of $h$

      What
      Use this activity to check your understanding of the effect of $h$ in quadratic equations. 

      How

      • Manipulate the sliders on the graph below to display $y=x^2$, $y=(x-2)^2$, and $y=(x+1)^2$. You should check your results using a table of values.
      • Then, describe the effect of $h$

      
      

    • answer (click to expand)

      $y =(x-h)^2$, the graph is shifted right $h$ units from the graph of $y = x^2$, and the vertex is $(h, 0)$ and in $y =(x+h)^2$, the graph is shifted left $h$ units from the graph of $y = x^2$, and the vertex is $(- h, 0)$.
      
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       Vertex Form of a Quadratic

      Applying all three transformations we have examined so far, we obtain a very useful form of the general quadratic: $y = a(x − h) ^2 + k$.
      
      Consider the case when $a = 1$. The smallest value of $y$ is $y = k$ which occurs when $(x − h) ^2 = 0$. We solve this to find that the minimum value occurs when $x = h$ and $y = k$. This is the coordinate $(h, k)$.
      
      The vertex form gets its name because we can immediately read the vertex from the quadratic: the vertex occurs at $(h, k)$. We use $a$ to tell us whether the vertex is a maximum or a minimum. 
      
    • Activity 6J - Vertex Form

      What
      This activity contains questions to check your knowledge of vertex form.

      How

      • Select the correct response to each of the multiple-choice questions.

      
      
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       Graphing Quadratic Functions

      To graph a quadratic function we should: 
      

      • Find the x-intercepts (if any). 
      • Find the y-intercept. 
      • Find the vertex. 
      • Determine if the quadratic opens up or down and determine if the vertex is a maximum or a minimum. 
      • Plot the intercepts and the vertex. Connect these points with a smooth curve.
    • Activity 6K - Graphing a Quadratic Function

      What
      Use this activity to check your understanding of graphing quadratic equations. 

      How

      • Manipulate the sliders to graph $y=(x-2)^2+3$.
      • Click the share icon () in the graphing tool and export an image of your completed graph.
      • Annotate your graph to show all of the important features.
      • Share your completed graph in the Padlet below by opening the Padlet link below and clicking the plus sign at the bottom right-hand side of the screen, then uploading the image of your finished graph.

      
      

      
      

      
      
      
      

    Mark as done
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    Activity 6G - Factored Form Interactive Content

    Mark as done
  • 

    Activity 6H - Factored Form Interactive Content

    Mark as done
  • 

    Activity 6J - Vertex Form Interactive Content

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