Question

Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -5 and 3 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?

Short answer

The value of \(a\) does not affect the x-intercepts or the axis of symmetry. It affects the y-coordinate of the vertex. The x-coordinate of the vertex is always the midpoint of the x-intercepts.

Step-by-step solution

Understanding the given form

The quadratic function can be written as \[ f(x)=a(x-r_{1})(x-r_{2}) \]where \( r_{1} \) and \( r_{2} \) are the x-intercepts.

Identify Intercepts

The x-intercepts are given as -5 and 3, so we have: \( r_{1} = -5 \) and \( r_{2} = 3 \).

Finding the quadratic functions for various values of a

- For \( a=1 \): \[ f(x) = 1(x + 5)(x - 3) = x^2 + 2x - 15 \]- For \( a=2 \): \[ f(x) = 2(x + 5)(x - 3) = 2x^2 + 4x - 30 \]- For \( a=-2 \): \[ f(x) = -2(x + 5)(x - 3) = -2x^2 - 4x + 30 \]- For \( a=5 \): \[ f(x) = 5(x + 5)(x - 3) = 5x^2 + 10x - 75 \]

Effect of a on intercepts

The value of \( a \) does not affect the intercepts. The intercepts depend solely on the values of \( r_{1} \) and \( r_{2} \).

Effect of a on the axis of symmetry

The axis of symmetry can be found using the formula \[ x = \frac{r_{1} + r_{2}}{2} = \frac{-5 + 3}{2} = -1 \]The axis of symmetry is \( x = -1 \), and is independent of the value of \( a \).

Effect of a on the vertex

The vertex of the parabola is located at the axis of symmetry. Since the y-coordinate of the vertex depends on \( a \), different values of \( a \) will produce different vertices along the line \( x = -1 \), but the \( x \)-coordinate remains constant.

Comparing x-coordinate of the vertex with midpoint of intercepts

The x-coordinate of the vertex, found at \( x = -1 \), is exactly the midpoint of the intercepts \(-5\) and \(3\). It shows that the vertex lies on the axis of symmetry calculated using the x-intercepts. This means that for any quadratic function, the vertex x-coordinate is the midpoint of the x-intercepts.

Key concepts

x-intercepts

The x-intercepts of a quadratic function, also known as the roots, are the points where the graph crosses the x-axis. These points are found by solving the equation set to zero. In our example with intercepts at -5 and 3, they are where the function equals zero:

\(r_1 = -5\) and \(r_2 = 3\).

To get the quadratic function from the x-intercepts and a coefficient \(a\), use:

\(f(x) = a(x - r_1)(x - r_2)\).

The x-intercepts determine the basic structure of the parabola, showing where it touches the x-axis. Importantly, the values of \(a\) do not change these intercepts. They only modify the shape of the parabola.

axis of symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. This line passes through the vertex (the highest or lowest point of the parabola).

For a quadratic function \(f(x) = a(x - r_1)(x - r_2)\), the axis of symmetry can be calculated as:

\(x = \frac{r_1 + r_2}{2}\).

In our example, substituting \(r_1 = -5\) and \(r_2 = 3\), the axis of symmetry is:

\(x = \frac{-5 + 3}{2} = -1\).

This remains constant no matter the value of \(a\). The axis of symmetry, therefore, depends only on the x-intercepts.

vertex

The vertex of a parabola is a crucial point since it represents the minimum or maximum value of the quadratic function. The vertex lies on the axis of symmetry.

For a quadratic function in the form \(f(x) = a(x - r_1)(x - r_2)\), its x-coordinate is the same as the axis of symmetry \(x = -1\) in our case.

To find the y-coordinate, we substitute \(x = -1\) back into the quadratic equation. Consequently, different values of \(a\) will change the y-coordinate:

- For \(a=1\), the vertex is at \((-1, f(-1)) = (-1, -16)\).
- For \(a=2\), the vertex is at \((-1, -32)\).
- For \(a=-2\), the vertex is at \((-1, 32)\).
- For \(a=5\), the vertex is at \((-1, -80)\).

While the x-coordinate remains the same, the y-coordinate varies with different values of \(a\).

parabola

A parabola is the graph of a quadratic function. Its shape is symmetric and it can open upwards or downwards depending on the value of \(a\):

- If \(a > 0\), the parabola opens upwards and the vertex is the minimum point.
- If \(a < 0\), the parabola opens downwards and the vertex is the maximum point.

In our case, adjusting the value of \(a\) changes the steepness of the parabola while keeping the x-intercepts and the axis of symmetry unchanged.

Having intercepts at -5 and 3 and varying \(a\), leads to different parabolas:

- For \(a=1\), it’s less steep compared to higher \(a\) values.
- Increasing \(a\) to 2 or 5 makes the curve steeper.
- A negative \(a = -2\) flips the parabola downwards.

Understanding these attributes help predict how changes in the quadratic function equation affect its graph.