Quadratic Graphs
Recognise that quadratic functions produce curved (parabolic) graphs, distinguish them from linear graphs, and use plotted quadratic graphs to estimate values and find approximate solutions
What a learner can do afterwards
- Explain why x² produces a U-shaped curve rather than a straight line
- Calculate and check coordinate pairs, use symmetry, and plot a quadratic from a table
- Read approximate solutions by locating a target height on a quadratic graph and reading the corresponding x-coordinates
- Describe vertical translations and find the constant in y = x² + c from a given point
- Use nonnegative squares to determine whether a quadratic can reach the x-axis
1 · Read
You have plotted straight lines from coordinate pairs. A quadratic rule has an x² term and no higher power of x. Its graph is a smooth curve called a parabola. For y = x², the graph opens upward in a U shape. The values grow as x moves away from zero in either direction.
Compare changes in y when x increases by equal steps. For y = x² at x = 1, 2, 3, 4, the y-values are 1, 4, 9, 16. The increases are 3, 5, 7. They are not constant. A straight line has the same change in y for every equal step in x.
The number multiplying x² decides the opening direction. A positive number gives an upward-opening parabola. A negative number gives a downward-opening parabola. For y = -x², square x first, then change the sign. Its highest point is (0, 0). A parabola's lowest or highest point is called its turning point. Other rules can also make curves, so a curved graph is not automatically quadratic.
To make a table, substitute each x into the rule. For y = x² - 6 at x = 2, square first: 2 × 2 = 4. Then subtract 6, giving y = -2. Record (2, -2). For a negative input, square the whole number: (-2)² = (-2) × (-2) = 4. Then carry out the subtraction.
A square is always zero or positive. For y = x² - 8, the smallest y-value is -8, at x = 0. The lowest point is (0, -8). There is no highest point when all real x-values are allowed. In a table of pairs, compare the y-values and keep each one matched with its x.
The x-axis is where y = 0. In a table, look for every pair whose y-coordinate is zero. For y = x² - 9, these pairs are (-3, 0) and (3, 0). They give two x-axis crossings. The negative x-value is the left crossing.
A negative input still has a nonnegative square. For y = x² - 1 at x = -2, use (-2)² = (-2) × (-2) = 4. Then y = 4 - 1 = 3. Writing -4 for the square would confuse squaring a negative number with putting a minus sign in front of a square.
You can find a crossing without a table for a rule such as y = x² - 49. Set y = 0, so x² must equal 49. Both 7 and -7 square to 49. The positive crossing is x = 7. This is an exact calculation; a crossing read from a drawn curve may only be an estimate.
To choose an equation from coordinate pairs, test each proposed rule. The point (2, 9) fits y = x² + 5 because 4 + 5 = 9. It does not fit y = x² - 5 because 4 - 5 = -1. Check every given point before choosing a rule.
For y = x² + c, opposite x-values have the same y-value because x² and (-x)² are equal. So a point (a, b) has a partner (-a, b). For example, (3, 14) and (-3, 14) both lie on y = x² + 5. This graph is symmetric about the y-axis.
The y-intercept is where x = 0. Substitute zero into the rule to find its y-value. For y = x² + 7, this gives y = 0² + 7 = 7, so the intercept is (0, 7). Do not confuse x = 0 at the y-axis with y = 0 at the x-axis.
To plot a quadratic, calculate several coordinate pairs on both sides of its turning point. Plot the pairs on axes with marked scales. Join them with a smooth curve, not straight segments. A table lists selected points; it does not show every point between them.
An upward-opening parabola need not cross the x-axis. For y = x² + 6, every y-value is at least 6 because x² is nonnegative. The graph cannot reach y = 0, so it has no x-axis crossings. Use the whole rule, not just its opening direction.
Read an estimate using the marked scale. If a curve's y-value lies halfway between the ticks for 4 and 5, estimate y as 4.5. A midpoint is half a tick interval above the lower tick. The line below shows that scale reading, not the curve.
Changing only the constant moves every y-value by the same amount. Changing y = x² - 2 to y = x² + 5 adds 7 to each y-value, so the whole graph moves up 7 units. The x-values stay the same.
If y = x² + c passes through (3, 11), use the point to find c. Substitute x = 3 and y = 11: 11 = 9 + c. Subtract 9, giving c = 2. Check by putting c back into the rule.
To estimate the x-values at a target height y = h, find where the plotted curve meets that height. From each meeting point, read straight down to the x-axis. Read every meeting, since a parabola may give two solutions. For an x-axis crossing, h = 0. If a crossing is one quarter of the way from x = 1 to x = 2, estimate x as 1.25. A written description of the crossing and scale can supply the same reading information.
Quadratic graphs are parabolas. Calculate and plot coordinate pairs, read scales carefully, and find x-axis crossings where y = 0. The sign of the x² coefficient decides the opening direction.
Additional questions and readings written for Marble.
2 · Watch
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Where this leads
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