Problem 40 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts and the vertex of the graph of the function. Then sketch the graph of the function. \(y=(x-4)(x+2)\)

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=(x-4)(x+2)\) are \(x=4\) and \(x=-2\), the vertex is at \((1, -9)\). The graph of the function is a parabola that opens upwards.

Step 1: Expand the function

Expand the given function - Apply the distributive property (FOIL - First, Outer, Inner, Last) to expand \(y=(x-4)(x+2)\), resulting in \(y=x^2-2x-8\).

Step 2: Find the x-intercepts

To find the x-intercepts, replace \(y\) with \(0\) and solve the resulting equation - \(0=x^2-2x-8\). Setting this equation to zero results in two solutions, \(x=4\) and \(x=-2\), which are the x-intercepts.

Step 3: Find the Vertex

The vertex of a parabola is located at \(x=\frac{-b}{2a}\). In our expanded equation \(y=x^2-2x-8\), \(a=1\) and \(b=-2\). Substituting these values into the vertex formula gives us the x-coordinate of the vertex as \(x=\frac{-(-2)}{2*1}=1\). Replacing \(x\) with \(1\) in the function \(y=(x-4)(x+2)\) will give us the y-coordinate of the vertex. Hence, the vertex is \((1,-9)\).

Step 4: Sketch the graph

Plot the x-intercepts at (4,0) and (-2,0), the vertex at (1,-9). Connect the points with a smooth curve that opens upwards since the coefficient of \(x^2\) in the expanded equation is positive.

Key Concepts

X-intercepts of Quadratic Functions

When it comes to understanding quadratic functions, it’s essential to know how to find their x-intercepts. These are the points where the graph of the quadratic function crosses the x-axis, essentially where the output value, or 'y', is zero.

To find the x-intercepts, you substitute 0 for 'y' in the quadratic equation and solve for 'x'. For instance, consider the function given in our exercise: \(y=(x-4)(x+2)\). By applying this principle and setting 'y' to 0, we have \(0=(x-4)(x+2)\). This equation can be solved by finding the values of 'x' that make each factor zero, which are \(x=4\) and \(x=-2\). These values represent the points where the parabola touches the x-axis, and they are crucial for graphing the function correctly.

Remember, the number of x-intercepts can vary; a quadratic function may have two, one, or no real x-intercepts, depending on whether the parabola crosses the x-axis in two places, just touches it, or does not touch it at all.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, known as the maximum or minimum of the function, respectively. It’s a fundamental characteristic of the parabola's shape and can be calculated using the formula \( x = \frac{-b}{2a} \).

For the equation \( y = x^2 - 2x - 8 \), 'a' and 'b' are coefficients from the standard form \( ax^2 + bx + c \). Here, \( a = 1 \) and \( b = -2 \). Plug these values into the vertex formula to find the x-coordinate of the vertex: \( x = \frac{-(-2)}{2*1} = 1 \). Then, substitute 'x' with 1 into the original function to find the y-coordinate: \( y = (1-4)(1+2) = -9 \). This gives us the vertex \( (1, -9) \), which is the turning point of the parabola.

Understanding the vertex is crucial for identifying the direction of the parabola (upwards or downwards) and its position relative to the x-axis. An upwards opening parabola has a minimum vertex, and a downwards opening one has a maximum vertex.

Quadratic Function Expansion

Expanding a quadratic function means to transform it from its factored form, such as \((x-4)(x+2)\), into its standard form, which looks like \(ax^2 + bx + c\). This process typically involves the use of the distributive property or FOIL (First, Outer, Inner, Last) method.

In our exercise, the expansion step was crucial to find both the x-intercepts and the vertex of the parabola. Starting with the factored form, \(y = (x - 4)(x + 2)\), we apply FOIL to get the expanded standard form: \(y = x^2 - 2x - 8\). Here’s a breakdown:

• First: Multiply the first terms, \(x \cdot x = x^2\).
• Outer: Multiply the outer terms, \(x \cdot 2 = 2x\).
• Inner: Multiply the inner terms, \(-4 \cdot x = -4x\).
• Last: Multiply the last terms, \(-4 \cdot 2 = -8\).

Combine the like terms, \(-2x\), from the Outer and Inner steps to complete the expansion. Understanding this process of expansion helps in graphing the function and determining properties such as the axis of symmetry and the direction of the parabola’s opening.