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2.04 Quadratic functions

Lesson

Key features

We've already looked at how to find certain features of linear equations including the gradient, the $x$x-intercept, and the $y$y-intercept. Now we are going to explore the features of parabolas.

 

Intercepts

Remember

The $x$x-intercepts are where the parabola crosses the $x$x-axis. This occurs when $y=0$y=0.

The $y$y-intercept is where the parabola crosses the $y$y-axis. This occurs when $x=0$x=0.

You can see them in the picture below. 

Recall from properties of the discriminant that we may have none, one or two $x$x-intercepts. Matching the number of solutions to the quadratic equation where $y=0$y=0.

Maximum and minimum values

Maximum or minimum values are also known as the turning points, and they are found at the vertex of the parabola.

Parabolas that are concave up have a minimum value. This means the $y$y-value will never go under a certain value.

Parabolas that are concave down have a maximum value. This means the $y$y-value will never go over a certain value.

 

Axis of symmetry

Maximum and minimum values occur on a parabola's axis of symmetry. This is the vertical line that evenly divides a parabola into two sides down the middle. Using the general form of the quadratic $y=ax^2+bx+c$y=ax2+bx+c, substitute the relevant values of $a$a and $b$b into:

The equation of the axis of symmetry

$x=\frac{-b}{2a}$x=b2a

Does it look familiar? This is part of the quadratic formula. It is the half-way point between the two solutions.

 

Gradient of a curve

We have learned about positive and negative gradients when we looked at straight lines. Generally, straight lines had only one or the other.

However, a parabola has a positive gradient in some places and negative gradient in others. The parabola's maximum or minimum value, at which the gradient is $0$0, is the division between the positive and negative gradient regions. Hence why it is also called the turning point.

Look at the picture below. See how one side of the parabola has a positive gradient, then there is a turning point with a zero gradient, then the other side of the parabola has a negative gradient?

We can see in this particular graph that the gradient is positive when $x<1$x<1 and we can say that the parabola is increasing for these values of $x$x. Similarly, the gradient is negative when $x>1$x>1, and for these values of $x$x the parabola is decreasing.

 

Practice questions

Question 1

Examine the given graph and answer the following questions.

Loading Graph...

A graph displaying a parabola opening upward on a Cartesian coordinate system with vertex located at$-2,-16$2,16. The x-axis is ranges from -10 to 5 and the y-axis ranges from -20 to 5. The parabola intercepts the x-axis at $\left(-6,0\right)$(6,0) and $\left(2,0\right)$(2,0), and y-axis at $\left(0,-12\right)$(0,12). The locations of the vertex and intercepts were not explicitly stated or mentioned.
  1. What are the $x$x values of the $x$x-intercepts of the graph? Write both answers on the same line separated by a comma.

  2. What is the $y$y value of the $y$y-intercept of the graph?

  3. What is the minimum value of the graph?

Question 2

Examine the attached graph and answer the following questions.

Loading Graph...
A parabola is drawn in a Cartesian cooridnate plane which opens downward. The vertex of the parabola has coordinates of ($-5$5,$0$0). The parabola intersects the y-axis at (0,$-25$25). The coordinates of the vertex and the y-intercept are not explicitly labeled or mentioned.

 

  1. What is the $x$x-value of the $x$x-intercept of the graph?

  2. What is the $y$y value of the $y$y-intercept of the graph?

  3. What is the absolute maximum of the graph?

  4. Determine the interval of $x$x in which the graph is increasing.

    Give your answer as an inequality.

Graphing quadratics

We can be expected to graph quadratic functions from any of the following forms:

Forms of quadratics

General Form: $y=ax^2+bx+c$y=ax2+bx+c 

Factored or $x$x-intercept form: $y=a\left(x-\alpha\right)\left(x-\beta\right)$y=a(xα)(xβ)

Turning point form: $y=a\left(x-h\right)^2+k$y=a(xh)2+k 

Each form has an advantage for different key features that can be identified quickly. However, for each form the role of $a$a remains the same.

That is:

  • If $a<0$a<0 then the quadratic is concave down
  • If $a>0$a>0 then the quadratic is concave up
  • The larger the magnitude of $a$a, the narrower the curve

 

Let's now look at examples for how to find all the key features for each form.

Graphing from turning point form

$y=a\left(x-h\right)^2+k$y=a(xh)2+k

  • This form is given this name as we can read the turning point directly from the equation: $(h,k)$(h,k)
  • We can obtain this graph from the graph of $y=x^2$y=x2, by dilating(stretching) the graph by a factor of $a$a from the $x$x-axis, then translating the graph $h$h units horizontally and $k$k units vertically
  • The axis of symmetry will be at $x=h$x=h
  • The $y$y-intercept can be found by substituting $x=0$x=0 into the equation
  • The $x$x-intercept can be found by substituting $y=0$y=0 into the equation and rearranging

 

Practice questions

Question 3

Consider the equation $y=-\left(x+2\right)^2+4$y=(x+2)2+4.

  1. Find the $x$x-intercepts. Write all solutions on the same line, separated by a comma.

  2. Find the $y$y-intercept.

  3. Determine the coordinates of the vertex.

    Vertex $=$=$\left(\editable{},\editable{}\right)$(,)

  4. Graph the equation.

    Loading Graph...

Question 4

Consider the equation $y=\left(x-3\right)^2+4$y=(x3)2+4.

  1. Does the graph have any $x$x-intercepts?

    No

    A

    Yes

    B
  2. Find the $y$y-intercept.

  3. Determine the coordinates of the vertex.

    Vertex $=$=$\left(\editable{},\editable{}\right)$(,)

  4. Graph the equation.

    Loading Graph...

 

Graphing from factored form

$y=a\left(x-\alpha\right)\left(x-\beta\right)$y=a(xα)(xβ)

  • From this form we can read the $x$x-intercepts directly: $(\alpha,0)$(α,0) and $(\beta,0)$(β,0). Since, if $y=0$y=0 we can solve the equation using the null factor law.
  • The axis of symmetry will be at the $x$x-coordinate midway between the two $x$x-intercepts. Alternatively you could expand this equation to general form and then use the formula for the axis of symmetry: $x=\frac{-b}{2a}$x=b2a.
  • The axis of symmetry is also the $x$x-coordinate of the turning point. Substitute this value into the equation the find the $y$y-coordinate of the turning point.
  • The $y$y-intercept can be found by substituting $x=0$x=0 into the equation

 

Practice question

Question 5

Consider the parabola $y=\left(x+1\right)\left(x-3\right)$y=(x+1)(x3).

  1. Find the $y$y value of the $y$y-intercept.

  2. Find the $x$x values of the $x$x-intercepts.

    Write all solutions on the same line separated by a comma.

  3. State the equation of the axis of symmetry.

  4. Find the coordinates of the vertex.

    Vertex $=$=$\left(\editable{},\editable{}\right)$(,)

  5. Graph the parabola.

    Loading Graph...

 

Graphing from general form

$y=ax^2+bx+c$y=ax2+bx+c

  • From this form we can read the $y$y-intercept directly: $(0,c)$(0,c) 
  • Find the axis of symmetry using the formula: $x=\frac{-b}{2a}$x=b2a
  • The axis of symmetry is also the $x$x-coordinate of the turning point. Substitute this value into the equation the find the $y$y-coordinate of the turning point.
  • The $x$x-intercept can be found by substituting $y=0$y=0 into the equation and then using one of our methods for solving quadratics, such as factorising or the quadratic formula, to find the zeros of the equation.

Alternatively, we can use the method of completing the square to rewrite the quadratic in turning point form.

 

Practice questions

Question 6

Consider the quadratic function $y=x^2+2x-8$y=x2+2x8.

  1. Determine the $x$x-value(s) of the $x$x-intercept(s) of this parabola. Write all answers on the same line separated by commas.

  2. Determine the $y$y-value of the $y$y-intercept for this parabola.

  3. Determine the equation of the vertical axis of symmetry for this parabola.

  4. Find the $y$y-coordinate of the vertex of the parabola.

  5. Draw a graph of the parabola $y=x^2+2x-8$y=x2+2x8.

    Loading Graph...

question 7

A parabola has the equation $y=x^2+4x-1$y=x2+4x1.

  1. Express the equation of the parabola in the form $y=\left(x-h\right)^2+k$y=(xh)2+k by completing the square.

  2. Find the $y$y-intercept of the curve.

  3. Find the vertex of the parabola.

    Vertex $=$= $\left(\editable{},\editable{}\right)$(,)

  4. Is the parabola concave up or down?

    Concave up

    A

    Concave down

    B
  5. Hence plot the curve $y=x^2+4x-1$y=x2+4x1

    Loading Graph...

Graphing quadratics using technology

Technology can be used to graph and find key features of a quadratic function. We may need to use our knowledge of the graph or calculate the location of some key features to find an appropriate view window.

Practice question

Question 8

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 $\left(\editable{},\editable{}\right)$(,)

Outcomes

1.2.2.1

examine examples of quadratically related variables

1.2.2.2

recognise and determine features of the graphs of 𝑦=𝑥^2,𝑦=𝑎𝑥^2 +𝑏x+𝑐, 𝑦=𝑎(𝑥−𝑏)^2+𝑐, and 𝑦=𝑎(𝑥−𝑏)(𝑥−𝑐), including their parabolic nature, turning points, axes of symmetry and intercepts

1.2.4.7

solve equations involving combinations of the functions above, using technology where appropriate

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