Problem 6 - Algebra 1: Concepts and Skills

Question

Does the graph of the function have x-intercepts of 4 and 5? \(y=4(x-4)(x-5)\)

Step-by-Step Solution

Answer

Yes, the function \(y=4(x-4)(x-5)\) does have x-intercepts at 4 and 5.

Step 1: Understand what x-intercepts mean

The x-intercepts of a function are those values of x for which the value of the function is zero. In other words, they are the points where the graph of the function intersects the x-axis.

Step 2: Set the function to zero

In order to find the x-intercepts, start by setting the equation equal to zero and solve for x. So the equation becomes: \(0=4(x-4)(x-5)\).

Step 3: Expand the equation

Next, expand the right side of the equation, which becomes: \(0=4x^2-36x+80\).

Step 4: Solve for x

The equation \(0=4x^2-36x+80\) is a quadractic equation and can be solved by factorization. The values of x that satisfy this equation are x=4 and x=5.

Step 5: Confirm the x-intercepts

Therefore, the function \(y=4(x-4)(x-5)\) does have x-intercepts at x=4 and x=5.

Key Concepts

Understanding Quadratic Functions

A quadratic function is a type of polynomial that can typically be written in the form \(y = ax^2 + bx + c\). In these functions, the highest degree or power of the variable \(x\) is 2. In our given function, \(y=4(x-4)(x-5)\), it is represented in a factored form which inherently makes it easier to identify the x-intercepts.

The main aspects of a quadratic function include:

• The vertex, which is the highest or lowest point on the graph depending on whether the parabola opens upwards or downwards.
• The axis of symmetry, which is a vertical line that passes through the vertex and divides the parabola into two mirror images. You can find this axis using the formula \(x = -\frac{b}{2a}\).
• The direction in which the parabola opens, determined by the sign of \(a\). If \(a > 0\), it opens upwards. If \(a < 0\), it opens downwards.

Understanding these characteristics helps in graphing the quadratic function and in solving it for specific values, such as finding the x-intercepts.

Factorization of Quadratic Functions

Factorization is a method where we express the quadratic function as a product of its linear factors. In simple terms, it involves rewriting the quadratic expression \(ax^2 + bx + c\) as \((x-p)(x-q)\) where \(p\) and \(q\) are the x-intercepts. This is exactly the transformation done in the given function, \(y = 4(x-4)(x-5)\), where we have identified the factors directly as \((x-4)\) and \((x-5)\).

This method is advantageous because:

• It allows easy identification of x-intercepts since these values are simply the solutions \(p\) and \(q\) when each factor is set to zero.
• It provides a quick way to sketch a graph since we know where the parabola will touch the x-axis.
• It simplifies solving quadratic equations, since solving the equation is reduced to solving simpler linear equations.

In our original problem, factorization is used as the primary technique to identify the x-intercepts, making it a crucial step in the solution.

Graph of a Quadratic Function

The graph of a quadratic function is a curve known as a parabola. For our function \(y=4(x-4)(x-5)\), after factorization, we easily recognize that it has x-intercepts at \(x=4\) and \(x=5\). This means the graph will touch or "intercept" the x-axis at these points.

Key characteristics of the graph include:

• Shape: The parabola is symmetric about its vertex, which is located directly between the x-intercepts. For this function, it lies at \(x = 4.5\), the midpoint between 4 and 5.
• Opening Direction: The coefficient of the squared term, \(a=4\), is positive, ensuring the parabola opens upwards.
• Width: Because \(a\) is larger than 1, the parabola will be narrower than the standard parabola \(y = x^2\).

This visual representation is paramount as it gives immediate insights into the behavior of the quadratic function. Understanding the graph facilitates easier prediction of the function’s behavior at various values of \(x\).