Key features
Previously, key features of linear equations have been explored including the gradient, the $x$x-intercept, and the $y$y-intercept. The graph of a quadratic equation is a parabola and is a curved or concave shape (either concave up or down, depending on the equation). The following sections describe the key features of a parabola.
Intercepts
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.
These intercepts are shown in the picture below:
Recall from properties of the discriminant that there may be one or two or even no $x$x-intercepts:
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.
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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 following equation:
$x=\frac{-b}{2a}$x=−b2a
Does it look familiar? This is part of the quadratic formula. It is the half-way point between the two solutions.
The gradient of a curve
Positive and negative gradients were previously explored for straight lines. 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. One side of the parabola has a positive gradient, there is a turning point with a zero gradient, and the other side of the parabola has a negative gradient.
In this particular graph, the gradient is positive when $x<1$x<1 and 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.
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?
Question 2
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.