1. Lesson 2

1.5. Explore

Mathematics 20-2 M3 Lesson 2

Module 3: Quadratics

 

Explore
 

You may have noticed when you were creating different graphs in the Discover section that the number of times the parabola touches or crosses the x-axis depends on whether the graph opens up or down and the location of the vertex (minimum or maximum value) of the graph.

 

This shows a graph with a parabola that opens upwards. Its vertex is a minimum value. This shows a graph with a parabola that opens downwards. Its vertex is a maximum value.

from: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 323. Reproduced by permission.

 

If the value of a is positive, the graph will open upwards. If the value of a is negative, the graph will open downwards. In each case, the position of the vertex will determine whether there are two, one, or zero x-intercepts.

 

If a is positive, the graph opens upwards and the following statements are true:

  • If the vertex is below the x-axis, there will be two x-intercepts because both arms of the parabola will cross the x-axis.

    This graph shows a parabola opening upwards with its vertex below the x-axis. The two x-intercepts are marked.

  • If the vertex is on the x-axis, there will be one x-intercept because the parabola touches the x-axis at one point.

    This graph shows a parabola opening upwards with its vertex on the x-axis. The single x-intercept is marked.

  • If the vertex is above the x-axis, there will be zero x-intercepts because the parabola will never cross the x-axis.

    This graph shows a parabola opening upwards with its vertex above the x-axis. There are no x-intercepts.

If a is negative, the graph opens downwards and the following statements are true:

 

  • If the vertex is below the x-axis, there will be zero x-intercepts because the parabola will never cross the x-axis.

    This graph shows a parabola opening downwards with its vertex below the x-axis. There are no x-intercepts.

  • If the vertex is on the x-axis, there will be one x-intercept because the parabola touches the x-axis at one point.

    This graph shows a parabola opening downwards with its vertex on the x-axis. There is one x-intercept marked.

  • If the vertex is above the x-axis, there will be two x-intercepts because both arms of the parabola will cross the x-axis.

    This graph shows a parabola opening downwards with its vertex above the x-axis. There are two x-intercepts marked.

If you are given the vertex and the direction of opening, can you decide how many x-intercepts the graph of the quadratic function will have?