Fit a curve to data: quadratic and other regressions

Enter paired data, choose a curved regression, read its coefficients, and estimate a value from the fitted equation.

6 steps

Your paired data rises and then falls, so a straight line cannot follow its shape. Fit a quadratic curve to these five pairs, then estimate $y$ when $x = 3.3$ :

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The quadratic form is $y = a + b x + c x^{2}$ . The same results list also offers logarithmic, exponential, power and inverse fits; see below for choosing those instead.

Enter the pairs, fit the quadratic, and estimate y

Below each step is the screen you should see after pressing.

Next to each step is the screen you should see after pressing.

Step 1

Open the mode menu and tap Statistics, then choose y=a+bx+cx². Start with an empty table; see below if old data is still there.

Press:
Step 2

In the first $x$ cell, enter $1, 2, 3, 4, 5$ , pressing the equals key after each value. The $y$ cells stay empty until you fill them.

Press:
Step 3

Tap the first $y$ cell. Enter $3, 6, 8, 8, 5$ , pressing the equals key after each value. The empty $y$ cell in row 6 is selected when you finish.

Press:
Step 4

Press the equals key on that empty cell, then tap Reg Results. In the $y = a + b x + c x^{2}$ block, read $a = - 2.8$ , $b = 6.6$ and $c = - 1$ . Check that it is marked In use.

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Step 5

Tap the top-left arrow of the results list to return to the table. Press the equals key on the empty row, then tap Statistics Calc. Its header should show $y = a + b x + c x^{2}$ .

Press:
Step 6

Enter $3.3$ , then open Statistics variables with the key shown. It opens at Regression, at the end of the sheet; tap $\overset{y}{^}$ . Press the equals key: the input is $3.3 \overset{y}{^}$ and the answer is $8.09$ .

Press:

Read and check the curve

Substitute the coefficients into $y = a + b x + c x^{2}$ :

y = - 2.8 + 6.6 x - x^{2}, \overset{y}{^} = - 2.8 + 6.6 (3.3) - (3.3)^{2} = 8.09.

Here $\overset{y}{^}$ means the estimated $y$ for the $x$ typed before it. At the original $x$ values $1, 2, 3, 4, 5$ , this curve gives $2.8, 6.4, 8, 7.6, 5.2$ : each is within $0.4$ of the entered $y$ .

Reg Results shows seven fits side by side and does not choose one for you. The quadratic has no $r$ or $R^{2}$ . Check predictions against your data before choosing a curve: the $y = a + b / x$ block, for example, predicts about $7.24$ at $x = 5$ , where your data has $y = 5$ . The In use badge tells you which fit the estimates use.

To find $x$ when $y = 7$ , enter $7$ in Statistics Calc, open the same variables list as in step 6, tap $\overset{x}{^}_{1}$ and press the equals key; then repeat with $\overset{x}{^}_{2}$ . For this curve the answers are approximately $2.25596934911$ and $4.34403065089$ ; the parabola reaches $y = 7$ twice. The screen groups long decimals in threes, so $2.255 96934911$ is $2.25596934911$ .

Other ways to estimate y

Under the seven blocks of Reg Results, the Estimate buttons ( $\overset{x}{^}_{1}$ , $\overset{x}{^}_{2}$ and $\overset{y}{^}$ for the quadratic) open Statistics Calc with that estimate already typed after an empty box: tap $\overset{y}{^}$ , enter $3.3$ in the box and press the equals key to get $8.09$ .

Choose another curved fit

To use another fit, choose its equation row when opening Statistics, or tap its block in Reg Results.

Each of these five fits shows $a$ , $b$ and $r$ . These are their data restrictions:

Fit	Equation	Required values
Logarithmic	$y = a + b ln (x)$	Every $x > 0$
Exponential	$y = a e^{b x}$	Every $y > 0$
Exponential, another form	$y = a b^{x}$	Every $y > 0$
Power	$y = a x^{b}$	Every $x > 0$ and $y > 0$
Inverse	$y = a + b / x$	Every $x \neq = 0$

For example, clear the table, then enter $x = 1, 2, 3, 4, 5$ and $y = 1.5, 2.9, 6.2, 12.1, 24.8$ as in steps 2 and 3. The values roughly double each step, which suggests an exponential curve. Choose $y = a e^{b x}$ as in step 1 (on the fx-991CW, tap its block in Reg Results); it gives $a = 0.7319663814$ , $b = 0.7039251806$ and $r = 0.9998096198$ . The $y = a b^{x}$ form gives the same curve with $b = 2.021672584$ . In Statistics Calc, entering $6 \overset{y}{^}$ as in step 6 gives $49.97537530963$ for both.

For curved fits, $r$ describes the data after transformation, such as taking logarithms. A value closer to $1$ in size means the transformed data lies closer to a straight line. Predictions beyond the entered $x$ range can differ greatly between fits. If your data follows a straight line, find a line of best fit instead.

If your result is different

Old numbers remain: tap the trash button on the function bar under the screen, then confirm with Delete All. This deletes the old data.
A $y$ cell is still empty: tap it and enter its value, as in step 3. A row without a $y$ is left out of every fit, so the coefficients change.
A block shows Math ERROR: the line under it says what the data needs; see the restrictions above. A $y$ of $0$ , for example, stops both exponential fits and the power fit. Two pairs are also insufficient for the quadratic: with just $(1, 3)$ and $(2, 6)$ , its block shows Math ERROR.
An estimate uses the wrong curve or shows Math ERROR: check the fit named in the Statistics Calc header; when that fit no longer works with your data, a line under it says what the data needs. Return to Reg Results and tap a block that shows coefficients. A block showing Math ERROR cannot be selected.
The results list covers the keys: tap its top-left arrow to return to the table.
A cell shows Syntax ERROR: press AC to clear the error.

The fx-991EX Statistics Calc screen showing 3.3ŷ with the estimated value 8.09.

Fit a curve to data on your fx-991EX calculator

Enter five pairs, fit y = −2.8 + 6.6x − x², and estimate y when x = 3.3.

Fit the curve More guides

Enter the pairs, fit the quadratic, and estimate y

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Read and check the curve

Other ways to estimate y

Choose another curved fit

If your result is different

Fit a curve to data on your fx-991EX calculator