Finding Unknown Coefficients Revision Notes for Leaving Cert Mathematics

Finding Unknown Coefficients

What are unknown coefficients?

Coefficients are the numerical values that appear in function equations. When we have a function like $f(x) = axe + b$ or $f(x) = ax^2 + bx + c$ , the letters $a$ , $b$ , and $c$ represent unknown coefficients that we need to find.

Finding unknown coefficients means determining these values when we're given information about specific points that lie on the function's graph. This is a fundamental skill in functions because it allows us to write the exact equation of a function from its graph or from given coordinate points.

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The key principle: if a point $(x, y)$ lies on the function, then substituting these coordinates into the function equation will give us a true mathematical statement.

Linear functions

A linear function has the form $f(x) = axe + b$ and represents a straight line when graphed. To find the unknown coefficients $a$ and $b$ , we need two pieces of information, typically two coordinate points that lie on the line.

For example, if $(2, 4)$ lies on the line, this means that when $x = 2$ , we get $f(2) = 4$ . This relationship allows us to create equations that we can solve to find the unknown values.

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Method for linear functions:

Substitute the first coordinate pair into $y = axe + b$ to create the first equation
Substitute the second coordinate pair into $y = axe + b$ to create the second equation
Solve the two equations simultaneously to find $a$ and $b$

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Worked Example: Finding coefficients of a linear function

Find the values of $a$ and $b$ for the function $y = axe + b$ that passes through the points $(-1, 2)$ and $(3, -2)$ .

Solution:

Since $(3, -2)$ lies on the line: $-2 = 3a + b$ ... (1)

Since $(-1, 2)$ lies on the line: $2 = -a + b$ ... (2)

Now we solve these simultaneously:

From equation (2): $2 = -a + b$ , so $a = b - 2$
Substitute into equation (1): $-2 = 3(b - 2) + b = 3b - 6 + b = 4b - 6$
Therefore: $4b = 4$ , so $b = 1$
From $a = b - 2$ : $a = 1 - 2 = -1$

Answer: $a = -1$ and $b = 1$ , so the function is $y = -x + 1$ .

Quadratic functions

A quadratic function contains a term in $x^2$ and has the general form $f(x) = ax^2 + bx + c$ . When graphed, quadratic functions create a smooth U-shaped curve called a parabola.

To find the three unknown coefficients $a$ , $b$ , and $c$ , we typically need three pieces of information, such as three coordinate points that lie on the parabola.

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Method for quadratic functions:

Substitute each coordinate pair into $y = ax^2 + bx + c$ to create three equations
Solve the system of three equations simultaneously
Often use elimination or substitution methods to find $a$ , $b$ , and $c$

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Worked Example: Finding coefficients of a quadratic function

The graph shows a quadratic function $f(x) = x^2 + bx + c$ . Find the values of $b$ and $c$ , and determine the coordinates of points $P$ and $Q$ .

Solution:

From the graph, we can see that $(-1, 0)$ and $(4, 5)$ lie on the curve.

Since $(-1, 0)$ lies on the curve: $f(-1) = 0$ $(-1)^2 + b(-1) + c = 0$ $1 - b + c = 0$ $-b + c = -1$ ... (1)

Since $(4, 5)$ lies on the curve: $f(4) = 5$ $(4)^2 + b(4) + c = 5$ $16 + 4b + c = 5$ $4b + c = -11$ ... (2)

Solving equations (1) and (2): From equation (1): $c = b - 1$ Substitute into equation (2): $4b + (b - 1) = -11$ $5b - 1 = -11$ $5b = -10$ $b = -2$

From $c = b - 1$ : $c = -2 - 1 = -3$

Therefore: $f(x) = x^2 - 2x - 3$

Finding point P (x-intercept): Set $f(x) = 0$ : $x^2 - 2x - 3 = 0$ Factoring: $(x - 3)(x + 1) = 0$ So $x = 3$ or $x = -1$ Since we already know $(-1, 0)$ , point $P$ is $(3, 0)$ .

Finding point Q (y-intercept): Set $x = 0$ : $f(0) = 0^2 - 2(0) - 3 = -3$ So point $Q$ is $(0, -3)$ .

Key steps for finding unknown coefficients

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Essential Steps:

Identify the function type - linear ( $ax + b$ ) or quadratic ( $ax^2 + bx + c$ )
Use given information - coordinates, intercepts, or other properties
Substitute coordinates - create equations by substituting known points
Solve simultaneously - use elimination or substitution methods
Check your answer - substitute back to verify the solution works

Exam tips

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Critical Exam Strategies:

Always substitute coordinates carefully - make sure you put the $x$ -value and $y$ -value in the correct positions
Label your equations - use (1), (2), (3) to keep track when solving simultaneously
Show clear working - demonstrate each step of the simultaneous equation solution
Check intercepts - use $x$ -intercepts and $y$ -intercepts to verify your final function
Read graphs carefully - identify all given coordinate points before starting

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Key Points to Remember:

Coefficients are the unknown numerical values in function equations that we need to determine
Linear functions need 2 pieces of information (usually 2 coordinate points) to find 2 unknowns
Quadratic functions need 3 pieces of information to find 3 unknowns
Substitute given coordinates into the function equation to create a system of equations
Solve simultaneously using elimination or substitution methods to find the coefficient values

Finding Unknown Coefficients (Leaving Cert Mathematics): Revision Notes