Question

Determine conditions on the constants \(a, b,\) and \(c\) such that the graph of \(f(x)=\frac{a x+b}{c x-a}\) is symmetric about the line \(y=x\).

Short answer

The graph of the function \(f(x)=\frac{a x+b}{c x-a}\) is symmetric about the line \(y=x\) if the constants \(a, b,\) and \(c\) obey the following condition: \(a^2 + bc = 0\).

Step-by-step solution

Setting up the equation

Take the function \(f(x)=\frac{a x+b}{c x-a}\) and set it equal to \(x\). This leads us to: \(x = \frac{a x+b}{c x-a}\).

Simplify the equation

To simplify this equation, first cross-multiply the equation to get rid of the fraction: \(x (c x - a) = a x + b\), which simplifies to: \(c x^2 - a x = a x + b\). Then, combine like terms: \(c x^2 - 2 a x - b = 0\).

Find the discriminant

For any quadratic equation in the form of \(ax^2 + bx + c = 0\), it's symmetry about the line \(y= x\) if the discriminant equates to zero. The discriminant is given by \(b^2 - 4ac\). Here, \(a = c\), \(b = -2 a\) and \(c = -b\). Therefore, the discriminant becomes: \((-2a)^2 - 4 * c * -b\).

Setting the discriminant to zero

For the graph to be symmetric about the line \(y = x\), the discriminant must be equal to zero. Therefore: \((-2a)^2 - 4 * c * -b = 0\), which simplifies to: \(4a^2 + 4bc = 0\). Dividing the entire equation by 4 yields: \(a^2 + bc = 0\).

Key concepts

Understanding the Discriminant of Quadratic Equations

When studying quadratic equations, an essential component to consider is the discriminant. The discriminant gives us valuable information about the nature of the roots of a quadratic equation without actually calculating the roots. It is represented by the formula \( b^2 - 4ac \) for the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).

For a graph to be symmetric about the line \( y=x \), the quadratic equation formed after setting \( f(x) \) equal to \( x \) must have roots that are mirror images of each other along this line. This can only happen if the roots are identical, which means the equation has a perfect square trinomial that factors into \( (x-h)^2 \), where \( h \) is the root's value. The requirement for a perfect square is that the discriminant must be equal to zero, \( b^2 - 4ac = 0 \).

Thus, if the discriminant is zero, the graph of the quadratic function will exhibit this line symmetry. In the case of \( f(x)=\dfrac{ax+b}{cx-a} \), the discriminant \( (-2a)^2 - 4 * c * -b \) must be set to zero to achieve the desired symmetry, leading to the condition \( a^2 + bc = 0 \) after simplifying.

The Role of Cross Multiplication in Simplifying Equations

Cross multiplication is a technique used to simplify equations, especially when dealing with fractions. It involves multiplying the numerator of one fraction by the denominator of the other and vice versa, effectively removing the fractions from the equation.

In the context of this exercise, cross multiplication is applied to the equation \( x = \dfrac{ax+b}{cx-a}\). By cross multiplying, we obtain \( x(cx-a) = ax + b \), which rearranges to the quadratic equation \( cx^2 - 2ax - b = 0 \).

Without cross multiplication, solving for the symmetry condition would be much more cumbersome, as handling fractions can be complex. This technique simplifies the problem to a standard quadratic form, allowing further analytical steps such as finding the discriminant. In a general case, cross multiplication is a multiplying shortcut that is particularly useful when you're solve for an unknown in proportion problems or comparing ratios.

Simplifying Algebraic Expressions to Reveal Symmetry

Simplifying algebraic expressions is a foundational skill in algebra that helps uncover underlying patterns or properties, such as symmetry. In the given function \( f(x)=\dfrac{ax+b}{cx-a} \), symmetry about the line \( y=x \) would mean that for any \( x \), its corresponding \( f(x) \) value would reflect over this line.

The process of simplification may involve combining like terms, factoring, or applying algebraic identities. In our case, after setting the function equal to \( x \) and cross multiplying, we combined like terms to arrive at the quadratic equation \( cx^2 - 2ax - b = 0 \). This is a crucial step as it allows us to identify the coefficients \( a \) and \( c \) of the quadratic equation, which are then used in the discriminant formula.

Simplification is not just about making expressions shorter or more aesthetically pleasing; it's about making them more understandable and workable so that we can apply further operations, like finding the discriminant in this case, to explore the properties of the function, such as symmetry.