Question

Your friend states that quadratic functions with the same x-intercepts have the same equations, vertex, and axis of symmetry. Is your friend correct? Explain your reasoning.

Short answer

Your friend is incorrect. Quadratic functions with the same x-intercepts do not necessarily have the same equations, vertex, or axis of symmetry. This is because the vertex and axis of symmetry are determined by the coefficients \(a\) and \(b\) in the equation \(ax^2 + bx + c = 0\), not by the x-intercepts.

Step-by-step solution

Understand Quadratic Function

A quadratic function is of the form \(f(x) = ax^2 + bx + c\). The roots or x-intercepts of the function are given by the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which solve the equation \(ax^2 + bx + c = 0\). If two quadratic functions have the same x-intercepts (or roots), it means they will both solve to the same values of \(x\). This does not however imply that they have the exact same equation, vertex, or axis of symmetry.

Compute Vertex and Axis of Symmetry

For a quadratic function \(f(x) = ax^2 + bx + c\), the vertex is given by \((- \frac{b}{2a}, f(- \frac{b}{2a}))\). The axis of symmetry is the x-coordinate of the vertex, which is \(x = -\frac{b}{2a}\). Notice that these two are determined by the coefficients \(a\) and \(b\) and not by the x-intercepts.

Pattern Comparison

Two quadratic functions can have the same roots but different coefficients \(a\), \(b\), and \(c\). Consequently, they can have the same x-intercepts but without the same vertex or axis of symmetry. Take for example, the functions \(f(x) = x^2 - 2x + 1\) and \(g(x) = 2x^2 - 4x + 2\). Both functions have the same roots (x=1), therefore the same x-intercepts, but different vertices and axis of symmetry.

Conclusion

From the pattern comparison, it's clear that your friend's statement is incorrect. While quadratic functions with the same roots do have the same x-intercepts, they do not necessarily have the same equation, vertex, or axis of symmetry.

Key concepts

X-Intercepts

X-intercepts, also known as roots or zeros of the quadratic function, are the points where the graph of the function crosses the x-axis. To find the x-intercepts of a quadratic function given by the equation \( f(x) = ax^2 + bx + c \), we solve the equation \( ax^2 + bx + c = 0 \). The solutions to this equation can be found using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The "\( \pm \)" sign in the formula means that there are generally two solutions, representing the x-intercepts.
Having the same x-intercepts means two different quadratic functions can intersect the x-axis at the same points. However, these functions can have distinctly different shapes or positions on the graph, depending on their coefficients.

Vertex

The vertex of a quadratic function is a significant point that represents the peak or the trough of the parabola, depending on whether it opens upwards or downwards. The standard vertex form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \). To find the vertex, we use the formula:

• \( ( -\frac{b}{2a}, f(-\frac{b}{2a}) ) \)

Here, \( -\frac{b}{2a} \) gives the x-coordinate of the vertex, and substituting this back into the function \( f(x) \) gives the y-coordinate.
Although two quadratic functions may share the same x-intercepts, they can still differ in their vertices due to differing coefficients. This means that even at the same x-intercepts, one parabola can be oriented differently or shifted vertically compared to another.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola and effectively divides the parabola into two symmetrical halves. This axis is a crucial part of understanding the symmetry and orientation of quadratic functions.
For a quadratic function described by \( f(x) = ax^2 + bx + c \), the formula to determine the axis of symmetry is:

• \( x = -\frac{b}{2a} \)

This line is pivotal because it reflects how the parabola stretches or compresses across that line.
Though different quadratic functions can have the same x-intercepts, their axes of symmetry may differ. This is largely because the axis of symmetry is dependent on the specific coefficient \( b \) and \( a \), not solely on the x-intercepts.

Quadratic Formula

The quadratic formula is a powerful tool used to determine the solutions to any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides the means to find the x-intercepts of a quadratic function without requiring factoring or completing the square.
By using the quadratic formula, we can solve for the two potential values of \( x \) that make the function equal to zero. Understanding this formula is essential in solving quadratic problems that do not factor easily. Moreover, while the x-intercepts obtained from the quadratic formula are crucial, they do not define other properties like vertex or axis of symmetry, which depend on the values of the coefficients \( a \), \( b \), and \( c \).