Enter the coefficient a and choose what to calculate (the mode). For the range of y or the average rate of change, enter the x-interval p and q. For the value of y, enter a value of x.
You can use decimals, negative numbers and fractions such as 1/2. The fields change automatically to match the mode.
Result and graph
Enter the coefficient a and the x-interval (or a value of x) in the fields on the left and press "Calculate". The result and the graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and graphs
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- For the quadratic function \(y = ax^{2}\), find the range of y for a domain of x, \(p \le x \le q\). The steps also explain the two cases, whether or not the domain includes 0 (when it does, the minimum or maximum is 0 at the vertex)
- Find the average rate of change as x goes from \(p\) to \(q\), both by the definition \(\dfrac{\Delta y}{\Delta x}\) (change in y ÷ change in x), with steps, and by the shortcut formula \(a(p+q)\)
- Find the value of y for a value of x (a point on the graph)
- The coefficient a can be a whole number, a decimal or a fraction such as 1/2
- The parabola is graphed. In range mode, the part for the domain and the range of y are highlighted; in rate-of-change mode, the line through the two points is drawn. A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page
This page covers \(y = ax^{2}\), whose vertex is at the origin. For the vertex and the maximum or minimum of the general form \(y = ax^{2} + bx + c\), use the related page "Completing the Square Calculator".
What is this calculation used for?
The path of a thrown ball
If you ignore air resistance, a ball thrown at an angle follows a parabola (the graph of \(y = ax^{2}\), shifted to another place).
A fly ball in baseball, a basketball shot and the water from a fountain all follow this shape, and this function is used to simulate sports and to move balls in video games.
A falling object's distance varies with the square of time (free fall)
If you ignore air resistance, the distance a dropped object falls is directly proportional to the square of the time since it was let go. With distance in feet and time in seconds, the constant is about 16, giving the quadratic function \(y = 16x^{2}\).
About 16 feet in 1 second, about 64 feet in 2 seconds: doubling the time makes the distance four times as large. This law, discovered by Galileo, is the starting point for the motion calculations in physics.
A car's braking distance grows with the square of its speed (road safety)
The distance a car travels from the moment the brakes are applied until it stops (the braking distance) is known to be roughly proportional to the square of its speed. Double the speed, and the braking distance becomes about four times as long.
This square relationship is the mathematical reason why speeding is dangerous. It is taught in driver's education, and it is a use of this function that directly affects safety.
Satellite dishes and reflectors (gathering radio waves or light at one point)
The shape you get by spinning a parabola around its axis (a paraboloid) has a special property: radio waves or light coming in parallel to the axis all meet at one point, the focus.
Satellite TV dishes and radio telescopes use this to collect weak radio waves. Flashlights and car headlights use it the other way around, to send light out in a straight beam.
Wind force grows with the square of wind speed (storms and buildings)
The force of the wind pushing on an object (wind pressure) is roughly proportional to the square of the wind speed. Double the wind speed, and the force becomes about four times as strong. This is why damage rises so quickly when a hurricane's winds get just a little faster.
When designing buildings and bridges, engineers use this square relationship to estimate the wind pressure from the expected wind speed, and decide how strong the structure must be.
Formulas and graphs
The quadratic function \(y = ax^{2}\)
Graph
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(y\)
\(=\)
\(a\)
\(\times\)
\(x^{2}\)
③
value of \(y\)
\(=\)
②
\(a\): coefficient
\(\times\)
①
\(x^{2}\): \(x\) squared
The formula in words
①
Take
\(x^{2}\): the value of \(x\) squared
②
multiply it by the
\(a\): coefficient
③
and you get the
value of \(y\)
Quick example
For \(y = 2x^{2}\), the value of \(y\) when \(x = 3\) is
value of \(y\)
\(=\)
coefficient (2)
\(\times\)
3 squared (9)
\(y = 2 \times 3^{2} = 2 \times 9 = 18\)
Key idea
\(y = ax^{2}\) is a function where "y varies directly with the square of x". It is direct variation \(y = ax\) with \(x\) replaced by \(x^{2}\), so \(a\) can also be called the constant of variation.
The graph is a smooth curve (a parabola) with its vertex at the origin (0, 0). If \(a > 0\), it opens upward; if \(a < 0\), it opens downward. The larger the absolute value of \(a\), the narrower (steeper) the parabola.
When you calculate, remember the order: square first, then multiply. To square a negative number, put it in parentheses, as in \(\left(-3\right)^{2} = 9\).
Range of y (when the domain includes \(0\), \(a > 0\))
Graph
Standard notation
(the usual math form)
\(0\)
\(\le\)
\(y\)
\(\le\)
\(am^{2}\)
In words
(symbols replaced with words)
②
\(0\): \(y\)-coordinate of the vertex (minimum)
\(\le\)
③
\(y\): values of \(y\) over the domain
\(\le\)
①
\(am^{2}\): value at the end \(m\) farther from \(0\) (maximum)
The formula in words
①
Let \(m\) be whichever end of the domain, \(p\) or \(q\), is farther from \(0\) (the one with the larger absolute value). The maximum is the
\(am^{2}\): value at the end \(m\) farther from \(0\)
②
the minimum is not an end value but the
\(0\): \(y\)-coordinate of the vertex
③
so the
\(y\): values of \(y\) over the domain
are greater than or equal to \(0\) and less than or equal to \(am^{2}\)
Quick example
For \(y = x^{2}\) with the domain \(-1 \le x \le 2\) (the end farther from \(0\) is \(m = 2\))
\(y\)-coordinate of the vertex (0)
\(\le\)
value of \(y\)
\(\le\)
value at \(x = 2\) (4)
\(x = -1 \Rightarrow y = 1, \quad x = 0 \Rightarrow y = 0, \quad x = 2 \Rightarrow y = 4\)
\(0 \le y \le 4\)
Key idea
Do not find the range of y just by substituting the end values of x. The right way is to picture the part of the parabola for the domain (the part where \(p \le x \le q\)) and read off its lowest and highest values of y.
When the domain includes \(x = 0\) (the vertex), the graph comes down to the vertex (0, 0) and turns back up, so the minimum is \(0\), not an end value. With the example domain \(-1 \le x \le 2\), a classic mistake is to look only at the end values \(1\) and \(4\) and answer "\(1 \le y \le 4\)".
When the domain does not include \(0\), the graph only increases or only decreases on that interval, so the end values are the minimum and the maximum.
When the two ends of the domain have the same absolute value (for example, \(-2 \le x \le 2\)), \(am^{2}\) is the same at both ends, so the maximum is reached at both ends, \(x = \pm 2\).
When \(a < 0\) (the parabola opens downward), everything flips: the maximum is \(0\) at the vertex, and the minimum is \(am^{2}\).
Average rate of change (as \(x\) goes from \(p\) to \(q\))
Graph
Standard notation
(the usual math form)
\(\dfrac{\Delta y}{\Delta x}\)
\(=\)
\(aq^{2} - ap^{2}\)
\(\div\)
\(q - p\)
In words
(symbols replaced with words)
③
average rate of change
\(=\)
①
\(\Delta y\): change in \(y\) (ending \(y\) − starting \(y\))
\(\div\)
②
\(\Delta x\): change in \(x\) (ending \(x\) − starting \(x\))
The formula in words
①
Take the
\(\Delta y\): change in \(y\) (the ending \(y\) minus the starting \(y\))
②
divide it by the
\(\Delta x\): change in \(x\) (the ending \(x\) minus the starting \(x\))
③
and you get the
average rate of change
Quick example
For \(y = 2x^{2}\) as x goes from \(1\) to \(3\) (\(y\) goes from \(2\) to \(18\))
average rate of change
\(=\)
change in y (18 − 2 = 16)
\(\div\)
change in x (3 − 1 = 2)
\(\dfrac{\Delta y}{\Delta x} = \dfrac{18 - 2}{3 - 1} = \dfrac{16}{2} = 8\)
Key idea
You find the average rate of change the same way as for a linear function: change in y ÷ change in x. But there is one big difference. The rate of change of a linear function \(y = ax + b\) is the slope \(a\) wherever you measure it, while the average rate of change of \(y = ax^{2}\) is not constant: it depends on where x starts and where it ends.
On the graph, the average rate of change is the slope of the straight line through the two points (a secant line). A parabola is a curve whose steepness changes from place to place, so when the two points change, the slope of the line changes too.
Shortcut formula for the average rate of change of \(y = ax^{2}\): \(a(p+q)\)
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(\dfrac{\Delta y}{\Delta x}\)
\(=\)
\(a\)
\(\times\)
\((p + q)\)
③
average rate of change
\(=\)
②
\(a\): coefficient
\(\times\)
①
\(p + q\): starting \(x\) + ending \(x\)
The formula in words
①
Take
\(p + q\): the starting \(x\) plus the ending \(x\)
②
multiply it by the
\(a\): coefficient
③
and you get the
average rate of change
Quick example
For \(y = 2x^{2}\) as x goes from \(1\) to \(3\)
average rate of change
\(=\)
coefficient (2)
\(\times\)
sum of 1 and 3 (4)
\(2 \times (1 + 3) = 2 \times 4 = 8\)
Key idea
This formula is a shortcut that works only for \(y = ax^{2}\) (vertex at the origin). The change in y, \(aq^{2} - ap^{2}\), factors as \(a(q + p)(q - p)\), so dividing by the change in x, \(q - p\), leaves just \(a(p + q)\).
Calculating from the definition (\(\Delta y \div \Delta x\)) always gives the same value. It is a good idea to learn the meaning through the definition first, and use the shortcut to check your work or save time. For the general form \(y = ax^{2} + bx + c\), the shortcut becomes \(a(p+q)+b\).
The graph of the quadratic function \(y = ax^{2}\) is a parabola with its vertex at the origin (0, 0). Do not find the range of y from the end values alone; look at the part of the graph for the domain. When the domain includes \(x = 0\), the minimum (the maximum if \(a < 0\)) is \(0\) at the vertex. Unlike a linear function, the average rate of change is not constant; as \(x\) goes from \(p\) to \(q\), it equals \(a(p+q)\).
Symbols and terms
Symbols
| \(a\) | a | The coefficient: the number multiplying \(x^{2}\). It decides which way the graph opens (up if \(a > 0\), down if \(a < 0\)) and how narrow it is. By custom, letters near the start of the alphabet, \(a,\ b,\ c\), stand for fixed numbers. |
| \(x,\ y\) | ex, why | Variables: letters for numbers that can take many values. By custom, letters near the end of the alphabet, \(x,\ y,\ z\), stand for unknown or changing numbers. On this page, choosing \(x\) determines \(y\). |
| \(p,\ q\) | pee, cue | The two ends of the x-interval: the left and right ends of the domain in range mode, and the starting and ending values of x in rate-of-change mode. On this page, \(p < q\) always (p is the smaller one). |
| \(m\) | em | A letter used on this page to explain the range: whichever end of the domain, \(p\) or \(q\), is farther from \(0\) (the one with the larger absolute value). The maximum (the minimum if \(a < 0\)) is reached at this end. When the two ends have the same absolute value, both ends give the same value. |
| \(\Delta x,\ \Delta y\) | delta x, delta y | The change in x and the change in y. \(\Delta\) (delta) is the Greek letter for D, the first letter of "difference", and it stands for "value after the change − value before". |
| \(\le\) | is less than or equal to | An inequality sign that includes the end value. \(-1 \le x \le 3\) says "x is greater than or equal to −1 and less than or equal to 3", so both −1 and 3 are included. Some countries write it with two lines, \(\leqq\), with the same meaning. |
| \(x^{2}\) | x squared | \(x\) multiplied by itself (\(x^{2} = x \times x\)). The small 2 at the upper right (the exponent) tells how many times \(x\) is used as a factor. To square a negative number, use parentheses, as in \(\left(-3\right)^{2} = 9\). |
Terms
| function | When each value of x gives exactly one value of y, we say "y is a function of x". \(y = ax^{2}\) is a function too: choose x, and exactly one y is determined. |
| quadratic function | A function where y is given by a degree-2 expression in x. The simplest form is \(y = ax^{2}\), with its vertex at the origin; the general form \(y = ax^{2} + bx + c\) lets the vertex move away from the origin. |
| coefficient | The number multiplying a variable, such as the \(a\) in \(y = ax^{2}\). Because \(y = ax^{2}\) says "y varies directly with the square of x", this \(a\) is also called the constant of variation, just as in direct variation \(y = ax\). |
| parabola | The smooth, symmetric curve that is the graph of \(y = ax^{2}\). It has the same shape as the path of an object thrown at an angle. |
| vertex | The turning point of a parabola. The vertex of \(y = ax^{2}\) is the origin (0, 0); it is the lowest point of the graph if \(a > 0\) and the highest point if \(a < 0\). |
| origin | The center of the coordinate plane, the point (0, 0), where the x-axis and the y-axis cross. It is labeled \(O\), the first letter of "origin". |
| domain and range | The set of values a variable can take, written with inequalities such as \(-1 \le x \le 3\). The domain is the set of values of x; the range is the set of values y takes over that domain. |
| average rate of change | The change in y divided by the change in x (\(\Delta y \div \Delta x\)). It tells how much y increases, on average, each time x increases by 1. It is constant for a linear function (the slope a), but for a quadratic function it depends on the interval. |
| change in x, change in y | The amount of change, found as "value after − value before". If the value goes down, the change is negative (for example, going from 18 to 2 is a change of −16). |
| opens upward | When \(a > 0\), the parabola has a valley shape. This is also called concave up. |
| opens downward | When \(a < 0\), the parabola has a hill shape. This is also called concave down. |
| linear function | A function of the form \(y = ax + b\) (its graph is a straight line). Its biggest difference from a quadratic function is that its rate of change is the same everywhere. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the fastest way forward.
| Positive and negative numbers (Grade 7) |
|
| Direct variation (Grade 7) |
|
| Coordinates and graphs (Grades 6–7) |
|
| Inequalities and intervals (Grades 6–8) |
|
| Linear functions and rate of change (Grade 8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the range of y for a domain of x
| Coefficient a | 2 |
| Left end of the domain p | -1 |
| Right end of the domain q | 3 |
| y at the left end | =B1*B2^2 |
| y at the right end | =B1*B3^2 |
| Minimum of y | =IF(AND(B2<=0,B3>=0),MIN(0,B4,B5),MIN(B4,B5)) |
| Maximum of y | =IF(AND(B2<=0,B3>=0),MAX(0,B4,B5),MAX(B4,B5)) |
Table to find the average rate of change
| Coefficient a | 2 |
| Starting x (p) | 1 |
| Ending x (q) | 3 |
| Starting y | =B1*B2^2 |
| Ending y | =B1*B3^2 |
| Average rate of change (Δy÷Δx) | =(B5-B4)/(B3-B2) |
| Check with a(p+q) | =B1*(B2+B3) |
Table to find y for a value of x
| Coefficient a | 2 |
| Value of x | 3 |
| Value of y | =B1*B2^2 |
After pasting, the upper rows (the coefficient and the x-interval) are your inputs, and the lower rows are calculated automatically. "^" raises to a power (^2 is squared), and "*" is multiplication.
The first table is y = 2x² with −1 ≤ x ≤ 3. The values of y at the left and right ends are 2 and 18, and the minimum 0 and maximum 18 are found (the range of y is 0 ≤ y ≤ 18). The IF(AND(B2<=0,B3>=0), …) part checks whether the domain includes 0; when it does, the 0 at the vertex is also a candidate for the minimum or maximum. It works as is when a is negative.
The second table is y = 2x² as x goes from 1 to 3. The definition cell (Δy÷Δx) and the shortcut cell a(p+q) both give 8.
The third table substitutes x = 3 into y = 2x², giving 18.
The first table is y = 2x² with −1 ≤ x ≤ 3. The values of y at the left and right ends are 2 and 18, and the minimum 0 and maximum 18 are found (the range of y is 0 ≤ y ≤ 18). The IF(AND(B2<=0,B3>=0), …) part checks whether the domain includes 0; when it does, the 0 at the vertex is also a candidate for the minimum or maximum. It works as is when a is negative.
The second table is y = 2x² as x goes from 1 to 3. The definition cell (Δy÷Δx) and the shortcut cell a(p+q) both give 8.
The third table substitutes x = 3 into y = 2x², giving 18.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the range of y for a domain of x
| Coefficient a | 2 |
| Left end of the domain p | -1 |
| Right end of the domain q | 3 |
| y at the left end | =B1*B2^2 |
| y at the right end | =B1*B3^2 |
| Minimum of y | =IF(AND(B2<=0,B3>=0),MIN(0,B4,B5),MIN(B4,B5)) |
| Maximum of y | =IF(AND(B2<=0,B3>=0),MAX(0,B4,B5),MAX(B4,B5)) |
Table to find the average rate of change
| Coefficient a | 2 |
| Starting x (p) | 1 |
| Ending x (q) | 3 |
| Starting y | =B1*B2^2 |
| Ending y | =B1*B3^2 |
| Average rate of change (Δy÷Δx) | =(B5-B4)/(B3-B2) |
| Check with a(p+q) | =B1*(B2+B3) |
Table to find y for a value of x
| Coefficient a | 2 |
| Value of x | 3 |
| Value of y | =B1*B2^2 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the coefficient and the x-interval with your own numbers.
How to calculate it in Python
from fractions import Fraction
coefficient_a = Fraction(2) # coefficient a (a fraction such as 1/2 can be written Fraction(1, 2))
left_end = Fraction(-1) # left end of the domain p (for the rate of change, the starting x)
right_end = Fraction(3) # right end of the domain q (for the rate of change, the ending x)
y_left = coefficient_a * left_end ** 2
y_right = coefficient_a * right_end ** 2
# Range of y: split into cases by whether the domain includes 0 (the vertex)
if left_end <= 0 <= right_end:
candidates = [Fraction(0), y_left, y_right]
else:
candidates = [y_left, y_right]
y_min = min(candidates)
y_max = max(candidates)
print(f"Range of y: {y_min} <= y <= {y_max}")
# Average rate of change (as x goes from p to q)
rate = (y_right - y_left) / (right_end - left_end)
print(f"Average rate of change (definition Δy÷Δx): {rate}")
print(f"Average rate of change (shortcut a(p+q)): {coefficient_a * (left_end + right_end)}")
The fractions module in the standard library lets you calculate with exact fractions, with no decimal rounding errors. This example is y = 2x² with the domain −1 ≤ x ≤ 3. When you run it, it shows the range of y, "0 <= y <= 18", and the average rate of change as x goes from −1 to 3, "4" (the same value by the definition and by the shortcut). Change the coefficient and the interval, and run it again.
How to write it in LaTeX and other math languages (copy and paste)
The quadratic function \(y = ax^{2}\)
y = ax²
y = ax^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>y</mi>
<mo>=</mo>
<mi>a</mi>
<msup><mi>x</mi><mn>2</mn></msup>
</mrow>
</math>
y = a x^2
a x^2
y := a*x^2;
y = a*x^2;
y = ax^2
Range of y (when the domain includes \(0\), \(a > 0\))
0 ≤ y ≤ am²
0 \le y \le am^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mn>0</mn>
<mo>≤</mo>
<mi>y</mi>
<mo>≤</mo>
<mi>a</mi>
<msup><mi>m</mi><mn>2</mn></msup>
</mrow>
</math>
0 <= y <= a m^2
{MinValue[{a x^2, p <= x <= q}, x], MaxValue[{a x^2, p <= x <= q}, x]}
minimize(a*x^2, x = p .. q); maximize(a*x^2, x = p .. q);
m = max(abs(p), abs(q)); ymin = 0; ymax = a*m^2;
0 ≤ y ≤ am^2
Average rate of change (as \(x\) goes from \(p\) to \(q\))
Δy/Δx = (aq² − ap²)/(q − p)
\dfrac{\Delta y}{\Delta x} = \dfrac{aq^{2} - ap^{2}}{q - p}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac>
<mrow><mi>Δ</mi><mi>y</mi></mrow>
<mrow><mi>Δ</mi><mi>x</mi></mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi><msup><mi>q</mi><mn>2</mn></msup>
<mo>−</mo>
<mi>a</mi><msup><mi>p</mi><mn>2</mn></msup>
</mrow>
<mrow><mi>q</mi><mo>−</mo><mi>p</mi></mrow>
</mfrac>
</mrow>
</math>
(Delta y)/(Delta x) = (a q^2 - a p^2)/(q - p)
(a q^2 - a p^2)/(q - p)
rate := (a*q^2 - a*p^2)/(q - p);
rate = (a*q^2 - a*p^2)/(q - p);
Δy/Δx = (aq^2 − ap^2)/(q − p)
Shortcut formula for the average rate of change of \(y = ax^{2}\): \(a(p+q)\)
Δy/Δx = a(p + q)
\dfrac{\Delta y}{\Delta x} = a(p + q)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac>
<mrow><mi>Δ</mi><mi>y</mi></mrow>
<mrow><mi>Δ</mi><mi>x</mi></mrow>
</mfrac>
<mo>=</mo>
<mi>a</mi>
<mo>(</mo>
<mi>p</mi>
<mo>+</mo>
<mi>q</mi>
<mo>)</mo>
</mrow>
</math>
(Delta y)/(Delta x) = a(p + q)
a (p + q)
rate := a*(p + q);
rate = a*(p + q);
Δy/Δx = a(p + q)
How to have ChatGPT do the calculation
You are a math calculation assistant for functions. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the quadratic function y = 2x², find the following two things. 1. The range of y when the domain of x is −1 ≤ x ≤ 3 (also explain why it matters whether the domain includes 0) 2. The average rate of change as x goes from 1 to 3 (show the steps for change in y ÷ change in x, and also use the shortcut formula a(p+q)) In Python, use the fractions module from the standard library to calculate exactly. Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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