Problem 24 - Precalculus : Building Concepts and Connections

Question

Factor to find the \(x\)-intercepts of the parabola described by the quadratic function. Also find the real zeros of the function. $$f(x)=6 x^{2}-x-2$$

Step-by-Step Solution

Answer

The x-intercepts of the parabola described by the function \(f(x)=6x^{2}-x-2\) are (-2/3, 0) and (1/2, 0). Also, the real zeros of this function are \(x=-2/3\) and \(x=1/2\).

Step 1: Factor the Quadratic Equation

Start by factoring the quadratic function \(f(x)=6x^{2}-x-2\). This function forms a quadratic equation of the form \(ax^{2}+bx+c=0\), where \(a=6\), \(b=-1\), and \(c=-2\). The quadratic equation can be factored by looking for two numbers that multiply to give \(ac = 6*(-2) = -12\) (the product of \(a\) and \(c\)) and that add to give \(b=-1\) (the coefficient of \(x\)). These two numbers are -3 and 4. Rewrite the equation as \(6x^{2}-3x+4x-2=0\). Then, factor by grouping: \(3x(2x-1) +2(2x-1)= 0\). This gives the factored form of the quadratic function as \((3x+2)(2x-1)=0\).

Step 2: Find the x-intercepts

The x-intercepts are points on the graph where \(y=0\). When \(y=0\), the factored equation becomes \(0= (3x+2)(2x-1)\). Using the zero-product property, we set each factor equal to zero and solve for \(x\). So, \(3x+2=0\) gives \(x=-2/3\) and \(2x-1=0\) gives \(x = 1/2\). So, the x-intercepts of the graph of this function are (-2/3, 0) and (1/2, 0).

Step 3: Find the real zeros of the function

The real zeros of a function are the x-coordinates where the function value is zero. These are the same as the x-intercepts for functions that cross the x-axis. So, in this case, the real zeros of the function are \(x=-2/3\) and \(x = 1/2\)

Key Concepts

x-intercepts

The x-intercepts of a quadratic function are critical points where the graph crosses the x-axis. These occur where the output or the y-value is zero. To find the x-intercepts for a function like \(f(x)=6x^{2}-x-2\), we need to set \(f(x)\) to zero and solve for \(x\).

This involves factoring the quadratic expression first. In our example, the function has been rewritten as \((3x+2)(2x-1)=0\). By setting each factor equal to zero, \(3x+2=0\) and \(2x-1=0\), we can solve for the x-values:

• For \(3x+2=0\), subtract 2 and divide by 3 to get \(x=-\frac{2}{3}\).
• For \(2x-1=0\), add 1 and divide by 2 to get \(x=\frac{1}{2}\).

These solutions, \((-\frac{2}{3}, 0)\) and \((\frac{1}{2}, 0)\), are the x-intercepts where the parabola meets the x-axis.

real zeros

Real zeros of a quadratic function are the x-values for which the function equals zero. These are also known as roots. They represent the solutions to the quadratic equation when set to zero.

For the function \(f(x)=6x^{2}-x-2\), after factoring we get \((3x+2)(2x-1)=0\). Solving each factor for zero, we find the real zeros:

• \(3x+2=0\) gives \(x=-\frac{2}{3}\).
• \(2x-1=0\) gives \(x=\frac{1}{2}\).

These real zeros \(-\frac{2}{3}\) and \(\frac{1}{2}\) are exactly where the graph crosses the x-axis, confirming the x-values at which the parabola's value is zero. For quadratic functions, real zeros are an important feature, providing insights into the graph's behavior.

quadratic functions

Quadratic functions are a fundamental component in algebra, defined by the standard form \(f(x) = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero. This type of function produces a parabola when graphed.

Key aspects of quadratic functions include:

• The coefficient \(a\) influences the direction of the parabola. If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• The vertex is the highest or lowest point of the parabola, providing valuable information about the function's minimum or maximum values.
• They can have zero, one, or two real zeros, which correspond to the x-values where the function equals zero.

In our example, \(f(x) = 6x^{2} - x - 2\), \(a = 6\), indicating an upward-opening parabola. Understanding the nature of quadratic functions helps solve equations and predict the graph's behavior over different values of \(x\). By factoring or using the quadratic formula, we can find the x-intercepts and real zeros efficiently.