Problem 44 - 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. \(y=(x-2)(x-6)\)

Step-by-Step Solution

Answer

The x-intercepts are at \(x=2\) and \(x=6\), and the vertex of the function is at \((4, -4)\).

Step 1: Find the x-Intercepts

Set the function equal to zero and solve for \(x\). Thus, we get: \(0=(x-2)(x-6)\). This gives us two solutions, \(x=2\) and \(x=6\), which are the x-intercepts of the function.

Step 2: Find the Vertex

The x-coordinate of the vertex can be found using the formula \(h = -b/2a\). In this expanded form \(y=a(x-h)^2+k\), our equation is \(y=x^2-8x+12\), i.e., \(a=1\), \(b=-8\) and \(c=12\). Therefore, \(h = -(-8)/2*1 = 4\). Substitute \(h=4\) into the equation to get the y-coordinate of the vertex: \(k = (4-2)(4-6) = -4\). So, the vertex of the function is at (4, -4).

Step 3: Sketch the Graph

Plot the x-intercepts and the vertex. Since the coefficient of \(x^2\) is positive, the parabola opens upwards. Now, sketch the graph based on this information.

Key Concepts

x-intercepts

To find the x-intercepts of a quadratic equation, we look for the points where the graph crosses the x-axis. These are the points where the value of y is zero. For the function given in the exercise, \( y = (x-2)(x-6) \), we set it equal to zero to solve for x. This can be written as:\[ 0 = (x-2)(x-6) \]To find the solutions, we set each factor to zero:

• \( x-2 = 0 \) which gives \( x = 2 \)
• \( x-6 = 0 \) which gives \( x = 6 \)

Thus, the x-intercepts are \( x = 2 \) and \( x = 6 \). These points tell us where the graph touches or crosses the x-axis, bridging our understanding of how a quadratic equation behaves on a coordinate plane. These intercepts are vital for sketching the graph accurately.

vertex of a parabola

The vertex of a parabola is a central point that acts as the peak or the lowest point of the graph, depending on whether it opens up or down. For the quadratic equation, this vertex can be calculated using the formula for the x-coordinate, \( h = -\frac{b}{2a} \), where \( a \) and \( b \) come from the standard form of the quadratic equation \( y = ax^2 + bx + c \).In our example, the quadratic form of \( y = (x-2)(x-6) \) expands to \( y = x^2 - 8x + 12 \):

• Here, \( a = 1 \) and \( b = -8 \).
• So, \( h = -\frac{-8}{2 \times 1} = 4 \).

To find the y-coordinate \( k \), substitute \( h = 4 \) back into the original equation:
\[ k = (4-2)(4-6) = -4 \]Thus, the vertex of the parabola is at the point \( (4, -4) \). Understanding the vertex helps determine the symmetrical axis of the parabola and its direction, which is essential for graph sketching.

sketching graphs

Sketching the graph of a quadratic function involves plotting key points that define its shape. For the function \( y = (x-2)(x-6) \), you already have the x-intercepts \( x = 2 \) and \( x = 6 \) and the vertex \( (4, -4) \). Use this information to draw the graph:

• Begin by plotting the x-intercepts and the vertex on a coordinate plane.
• Remember, the parabola will open upwards because the coefficient of \( x^2 \) is positive.
• Draw a smooth curve through the intercepts and the vertex, ensuring the vertex forms the lowest point of the curve.
• The parabola will be symmetrical around the vertical line passing through the vertex, which is \( x = 4 \).

By doing this, you get a visual representation of the quadratic function, allowing for a clearer understanding of its behavior. Sketching graphs is a useful skill as it combines algebraic and geometric perspectives, enhancing comprehension.