Question

Write a function \(g\) in standard form whose graph has the same \(x\)-intercepts as the graph of \(f(x)=2 x^2+8 x+2\). Find the zeros of each function by completing the square. Graph each function.

Short answer

The function \(g\) in standard form with the same x-intercepts as \(f(x) = 2x^2 + 8x + 2\) is \(g(x) = x^2 + 4x\). The x-intercepts for both functions are x = 0 and x = -4.

Step-by-step solution

Expression of Function f

Consider the given function \(f(x) = 2x^2 + 8x + 2\)

Find x-intercepts of Function f

To find the x-intercept(s) of this function, replace \(f(x)\) with 0 and solve for \(x\). \n Divide all the terms by 2, which gives \(x^2 + 4x + 1 = 0\). \n Complete the square on the quadratic equation to find the zeros: \((x + 2)^2 - 4 = 0\). \n Then solve for \(x\): \(x + 2 = \pm 2\), which gives \(x = 0, -4\).

Expression of Function g

Since g is supposed to have the same x-intercepts as f, the roots of g will also be 0, -4. A quadratic function with roots \(x_1\) and \(x_2\) is represented as \(g(x) = a(x - x_1)(x - x_2)\). Replacing \(x_1\) and \(x_2\) with the roots, gives \(g(x) = a(x - 0)(x - (-4)) = ax(x + 4)\). In order for it to be in standard form (like \(f(x) = 2x^2 + 8x + 2\)), we assume \(a = 1\) to match coefficients, so our final function g is \(g(x) = x(x + 4) = x^2 + 4x\)

Graph Function g

To graph \(g(x) = x^2 + 4x\), plot the x-intercepts (0,0) and (-4,0), and graph the function. The graph will be a parabola opening upwards

Key concepts

Standard Form of a Quadratic Function

Understanding the standard form of a quadratic function is crucial in the realm of algebra. It's written as \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a \) is not zero, as it defines the degree of the polynomial. For the function in our exercise, \( 2x^2 + 8x + 2 \) is in standard form with \( a = 2, b = 8, \) and \( c = 2 \).

Why is this form helpful? It allows for a straightforward identification of the parabola's direction (upward if \( a > 0 \) or downward if \( a < 0 \) ) and helps with finding key features like the vertex, axis of symmetry, and y-intercept. To convert other forms to standard, techniques like completing the square or expansion of factored forms are used.

X-Intercepts of a Quadratic Function

X-intercepts of a quadratic function are the points where the graph crosses the x-axis, representing the function's real roots. They're found by setting \( f(x) = 0 \) and solving for \( x \). In our exercise, the x-intercepts were found by completing the square, leading to \( x = 0 \) and \( x = -4 \).

These intercepts give valuable information when sketching the graph or analyzing the function's behavior. They indicate where the function changes from positive to negative values or vice versa. If a function lacks real x-intercepts, it means the parabola does not touch the x-axis at all.

Graphing Quadratic Functions

Graphing quadratic functions is a way to visually represent the function's characteristics. To graph such a function, identify the vertex, x-intercepts, and whether the parabola opens up or down. This is all derived from the standard form equation. For \( g(x) = x^2 + 4x \), you start by plotting the x-intercepts at (0, 0) and (-4, 0), as these are the solution to \( g(x) = 0\).

Next, you can plot the vertex, which is found by using the formula \( -b/2a \) for the x-coordinate. The y-coordinate can be found by substituting this x back into the function. Remember, since \( a > 0 \) here, the parabola opens upward. As you plot more points on either side of the vertex and connect them, the shape of the parabola becomes clear.

Finding Zeros of Quadratic Functions

The term zeros of quadratic functions, which are also the x-intercepts, refer to the values where \( f(x) = 0\). These are the points where the graph meets the x-axis. There are several methods to find zeros, such as factoring, using the quadratic formula, or completing the square, which is showcased in the given exercise.

Completing the square transforms a quadratic into a perfect square trinomial plus or minus a constant. For our example, \( f(x) = 2x^2 + 8x + 2 \), by dividing all terms by 2 and rearranging, we got a perfect square on the left and a constant on the right, leading us to the roots \( x = 0 \) and \( x = -4 \). Finding zeros is significant because it helps us not only graph the function accurately but also solve related optimization problems, such as finding maximum or minimum values.