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.
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 that we may have none, one or two $x$x-intercepts depending on the number of solutions to the quadratic equation where $y=0$y=0.
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. |
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:
$x=\frac{-b}{2a}$x=−b2a
Does it look familiar? This is part of the quadratic formula. It is the half-way point between the two solutions. Note that the vertex always sits on the axis of symmetry so when we find the equation of the axis of symmetry we are also finding the $x$x value of the vertex.
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.
Examine the given graph and answer the following questions.
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.
What is the $y$y value of the $y$y-intercept of the graph?
What is the minimum value of the graph?
Examine the attached graph and answer the following questions.
What is the $x$x-value of the $x$x-intercept of the graph?
What is the $y$y value of the $y$y-intercept of the graph?
What is the absolute maximum of the graph?
Determine the interval of $x$x in which the graph is increasing.
Give your answer as an inequality.
We can be expected to graph quadratic functions from any of the following forms:
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(x−α)(x−β)
Turning point form: $y=a\left(x-h\right)^2+k$y=a(x−h)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:
Let's now look at examples for how to find all the key features for each form.
$y=a\left(x-h\right)^2+k$y=a(x−h)2+k
Consider the equation $y=25-\left(x+2\right)^2$y=25−(x+2)2.
Does $y$y have a minimum or maximum value?
Maximum
Minimum
What is the maximum value of $y$y?
Consider the equation $y=-\left(x+2\right)^2+4$y=−(x+2)2+4.
Find the $x$x-intercepts. Write all solutions on the same line, separated by a comma.
Find the $y$y-intercept.
Determine the coordinates of the vertex.
Vertex $=$=$\left(\editable{},\editable{}\right)$(,)
Graph the equation.
$y=a\left(x-\alpha\right)\left(x-\beta\right)$y=a(x−α)(x−β)
Consider the parabola $y=\left(x+1\right)\left(x-3\right)$y=(x+1)(x−3).
Find the $y$y value of the $y$y-intercept.
Find the $x$x values of the $x$x-intercepts.
Write all solutions on the same line separated by a comma.
State the equation of the axis of symmetry.
Find the coordinates of the vertex.
Vertex $=$=$\left(\editable{},\editable{}\right)$(,)
Graph the parabola.
$y=ax^2+bx+c$y=ax2+bx+c
Alternatively, we can use the method of completing the square to rewrite the quadratic in turning point form.
Consider the quadratic function $y=x^2+2x-8$y=x2+2x−8.
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.
Determine the $y$y-value of the $y$y-intercept for this parabola.
Determine the equation of the vertical axis of symmetry for this parabola.
Find the $y$y-coordinate of the vertex of the parabola.
Draw a graph of the parabola $y=x^2+2x-8$y=x2+2x−8.
A parabola has the equation $y=x^2+4x-1$y=x2+4x−1.
Express the equation of the parabola in the form $y=\left(x-h\right)^2+k$y=(x−h)2+k by completing the square.
Find the $y$y-intercept of the curve.
Find the vertex of the parabola.
Vertex $=$= $\left(\editable{},\editable{}\right)$(,)
Is the parabola concave up or down?
Concave up
Concave down
Hence plot the curve $y=x^2+4x-1$y=x2+4x−1
Technology can be used to graph and find key features of a quadratic function. We may need to use our knowledge of the graph's domain and range or calculate the location of some key features to find an appropriate view window.
Use your calculator or other handheld technology to graph $y=4x^2-64x+263$y=4x2−64x+263.
Then answer the following questions.
What is the vertex of the graph?
Give your answer in coordinate form.
The vertex is $\left(\editable{},\editable{}\right)$(,)
What is the $y$y-intercept?
Give your answer in coordinate form.
The $y$y-intercept is $\left(\editable{},\editable{}\right)$(,)