062plottingparabolas

 Year 9 into 10 Step Up Program


 * Plotting Parabolas **toc

We are going to produce a parabola from a table of values

** Question 1 **
Table of Values

Use the rule //y = x 2 // to complete the following table: WARNING Remember that a negative times a negative gives a positive answer ( –3 × –3 = + 9)

Draw Graph
 * plot the points onto a set of axes
 * don't forget to label the axes with **X** and **Y**
 * join with a __smooth curve__ to produce a parabola
 * continue the curve out to the edge of the graphing area
 * label the parabola with the equation

Notice that the sides of the parabola are never vertical -- they keep going out to the sides.
 * There will be a point on the parabola for __every__ possible value of x.
 * If we "zoom out" to see a wider range of x-values it will still look like the parabolas shown here
 * Your parabolas should never look like a capital "U"

Analysis
 * state the nature of the turning point
 * state the coordinates of the turning point
 * state the coordinates of the y-intercept
 * state the coordinates of the x-intercepts
 * state the equation of the axis of symmetry

** Solution 1 **
Table of Values for y = x 2 :

Graph:



Analysis:

Nature of tp: Minimum Coords of tp: (0, 0) Y-Intercept: (0, 0) X-Intercept: (0, 0) Axis of symmetry: x = 0

** Question 2 **
Table of Values

Use the rule //y = –x 2 + 3// to complete the following table:

WARNING Remember that //–x 2 // means square x first and __then__ make it negative ( –3 2 = –9) This is different from (–3) 2 = +9

Draw Graph
 * plot the points onto a set of axes
 * don't forget to label the axes with **X** and **Y**
 * join with a __smooth curve__ to produce a parabola
 * extend the curve out to the edge of the graphing area
 * label the parabola with the equation

Analysis
 * state the nature of the turning point
 * state the coordinates of the turning point
 * state the coordinates of the y-intercept
 * state the coordinates of the x-intercepts (estimate)
 * state the equation of the axis of symmetry

** Solution 2 **
Table of Values fo y = –x 2 + 3 :

Graph:

Analysis:

Nature of tp: Maximum Coords of tp: (0, 3) Y-Intercept: (0, 3) X-Intercept: (–1.7, 0) and (1.7, 0) {estimates} Axis of symmetry: x = 0

** Question 3 **
Table of Values

Use the rule //y = 2x 2 – 6x// to complete the following table:

Draw Graph
 * when a coordinate gets too big to fit on the graph, eg (–2, 20), just leave it out
 * plot the points onto a set of axes
 * don't forget to label the axes with **X** and **Y**
 * join with a __smooth curve__ to produce a parabola
 * extend the curve out to the edge of the graphing area
 * label the parabola with the equation

Analysis
 * state the nature of the turning point
 * state the coordinates of the turning point (estimate or calculate)
 * state the coordinates of the y-intercept
 * state the coordinates of the x-intercepts
 * state the equation of the axis of symmetry

** Solution 3 **
Table of Values for y = 2x 2 – 6x :

Graph: Analysis:

Nature of tp: Minimum Coords of tp: (1.5, –4.5) {see to the right for tp} Y-Intercept: (0, 0) X-Intercepts: (0, 0) and (3, 0) Axis of symmetry x = 1.5

** Question 4 **
Table of Values

Use the rule //y = x 2 – 2x + 2// to complete the following table:

Draw Graph
 * plot the points onto a set of axes
 * don't forget to label the axes with **X** and **Y**
 * join with a __smooth curve__ to produce a parabola
 * extend the curve out to the edge of the graphing area
 * label the parabola with the equation

Analysis
 * state the nature of the turning point
 * state the coordinates of the turning point
 * state the coordinates of the y-intercept
 * state the coordinates of the x-intercepts
 * state the equation of the axis of symmetry

** Solution 4 **
Table of Values for y = x 2 – 2x + 2 :

Graph:

Analysis:

Nature of tp: Minimum Coords of tp: (1, 1) Y-Intercept: (0, 2) X-Intercept: none Axis of symmetry: x = 1

Summary

 * Any quadratic (with an x 2 term) produces a parabola.
 * Parabolas have a **__turning point__** which is either a __**minimum point**__ or a __**maximum point**__ on the graph.
 * Parabolas are __**symmetrical**__ with a vertical __**axis of symmetry**__ through the turning point

Notice that the sides of the parabola are never vertical -- they keep going out to the sides.
 * There will be a point on the parabola for __every__ possible value of x.
 * If we "zoom out" to see a wider range of x-values it will still look like the parabolas shown here
 * Your parabolas should never look like a capital "U"

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