Problem 65 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ y=4 x^{2}-16 x+17 $$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at \((2,1)\).

Step 1: Identify the Type of Equation

The equation given is in the form of a quadratic, evidenced by the presence of the squared term, and it is specifically a quadratic in terms of the variable \(x\). This indicates that it is a parabola.

Step 2: Adjust to Vertex Form

Rearrange the equation \(y=4x^2-16x+17\) to its vertex form \(y=a(x-h)^2+k\). To do this, complete the square:First, factor out the 4 from the quadratic terms: \[ y=4(x^2-4x)+17 \]Next, complete the square inside the parenthesis. Take the coefficient of \(x\), which is -4, divide by 2, giving -2, and square it to get 4. Add and subtract 4 inside the parenthesis:\[ y=4(x^2-4x+4-4)+17 \]

Step 3: Simplify and Finish the Square

Rewrite the quadratic expression as a perfect square and simplify:\[ y=4((x-2)^2-4)+17 \]Distribute the 4:\[ y=4(x-2)^2-16+17 \]Combine like terms:\[ y=4(x-2)^2+1 \].This expression is now in the vertex form \(y=a(x-h)^2+k\) with \(a=4\), \(h=2\), and \(k=1\). So the vertex is \((2, 1)\).

Step 4: Graph the Parabola

Plot the vertex of the parabola at \((2, 1)\). Since \(a = 4 > 0\), the parabola opens upwards and is vertically stretched compared to \(y=x^2\). Draw the graph accordingly.

Key Concepts

Parabola Vertex

In the context of quadratic equations, the vertex of a parabola is a particularly crucial point. It represents the minimum or maximum point of the curve, depending on the direction in which the parabola opens. For the equation in vertex form, given by \[ y = a(x-h)^2 + k \]the vertex \((h, k)\) provides the precise location on the graph. In our exercise, after rearranging to vertex form, we determined that the vertex is at the coordinates \((2, 1)\). This point is where the parabola achieves its lowest point, as the coefficient \(a\)is positive, indicating the parabola opens upwards. Identifying the vertex is essential for correctly graphing quadratic equations as it helps in understanding the parabola's symmetry and overall shape.

Completing the Square

'Completing the square' is a technique used to transform quadratic equations into a more manageable form, known as the vertex form. This strategy reveals the vertex of the parabola directly. Let's walk through this technique using the equation \(y=4x^2-16x+17\).

Start by factoring out the coefficient of the squared term from the quadratic and linear terms:

• Factor out \(4\) to get: \(y=4(x^2-4x)+17\).
• Next, find a value that completes the square. Halve the \(-4\) to get \(-2\), and then square it to \(4\).
• Add and subtract this square inside the bracket to maintain equivalence: \(y=4(x^2-4x+4-4)+17\).

This leads to an expression that simplifies the quadratic into a perfect square trinomial, helping to rewrite it as \((x-2)^2\). Completing the square efficiently reconfigures the quadratic expression, crucial for sketching a parabola's graph.

Graphing Parabolas

Graphing a parabola requires an understanding of its key features, including its vertex, direction, and width. To graph the equation \( y = 4(x-2)^2 + 1 \), use the vertex ((2, 1)) as the starting point.

Here's a simple guide:

• Direction: Since the coefficient \(a = 4\) is positive, the parabola opens upwards. This means that as you move away from the vertex, the graph extends upwards.
  
• Width: The value of \(a\) also indicates the parabola's "width" or "steepness". A larger absolute value of \(a\) results in a narrower curve. In comparison to \(y = x^2\), \( y = 4(x-2)^2 + 1 \) is vertically stretched.
  
• Symmetry: Parabolas are symmetrical. Once the vertex is plotted, you can plot additional points equidistant from the vertex on either side to accurately shape the curve.
  

Graphing requires carefully plotting these aspects to accurately represent the parabola's structure on the coordinate plane.