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Magazine Chapter 13: Problem 67
Write each equation in standard form, if it is not alreacty 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. $$ y=4 x^{2}-32 x+63 $$
Short Answer
Expert verified
The equation represents a parabola with vertex at \((4, -1)\).
Step by step solution
01
Recognize the Equation Type
The equation is a quadratic equation in terms of \(x\) and is not centered around the \(y^2\) term. Therefore, it represents a parabola.
02
Rewrite in Vertex Form
To convert to the standard form of a parabola, we begin by completing the square. The expression \(4x^2 - 32x + 63\) can be rearranged: \(4(x^2 - 8x) + 63\).
03
Complete the Square
To complete the square for \(x^2 - 8x\), take half of \(-8\) (which is \(-4\)), square it to get \(16\), and add and subtract \(16\) inside the parentheses: \(4((x^2 - 8x + 16) - 16) + 63\).
04
Simplify the Equation
The expression simplifies to \(4((x - 4)^2 - 16) + 63\). Expanding this gives \(4(x - 4)^2 - 64 + 63\), which results in \(4(x - 4)^2 - 1\).
05
Identify Vertex Form and Vertex Coordinates
The equation is now in the vertex form \(y = 4(x - 4)^2 - 1\). Thus, the vertex of the parabola is at \((4, -1)\).
06
Graph the Parabola
To graph the parabola, plot the vertex at \((4, -1)\) and note that the parabola opens upwards since the coefficient of \((x-4)^2\) is positive and is scaled by a factor of 4.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic equation
A quadratic equation is a polynomial equation of degree 2. Its general form is given by \(ax^2 + bx + c = 0\). When graphed, it forms a curve known as a **parabola**. The values of \(a\), \(b\), and \(c\) determine the shape and position of the parabola on the coordinate plane.
Some characteristics of a quadratic equation include:
Some characteristics of a quadratic equation include:
- It's a second-degree polynomial, meaning the highest power of the variable \(x\) is 2.
- The graph of a quadratic equation is a symmetric curve.
- It can open upwards or downwards depending on the sign of the leading coefficient \(a\).
Vertex form
The vertex form of a quadratic equation is a way of expressing the equation so that the vertex of the parabola is easily identifiable. This form is:
\(y = a(x-h)^2 + k\), where \((h, k)\) are the coordinates of the vertex. The conversion from standard form (\(ax^2 + bx + c\)) to vertex form is useful in graphing because:
\(y = a(x-h)^2 + k\), where \((h, k)\) are the coordinates of the vertex. The conversion from standard form (\(ax^2 + bx + c\)) to vertex form is useful in graphing because:
- It directly provides the vertex, thus indicating the highest or lowest point on the graph.
- The value of \(a\) still determines the direction and width of the parabola. Positive values of \(a\) mean the parabola opens upwards, while negative values mean it opens downwards.
- The vertex form makes it easier to sketch the graph by showing symmetry about the line \(x = h\).
Completing the square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This process makes it easier to find the vertex of the corresponding quadratic function.
Here鈥檚 how to complete the square:
Here鈥檚 how to complete the square:
- Start with the quadratic in the form \(ax^2 + bx + c\).
- Factor out the coefficient of \(x^2\) if it鈥檚 not 1 (as shown in the original solution, \(4(x^2 - 8x) + 63\)).
- Take half of the coefficient of \(x\), square it, and add and subtract this square inside the equation. For \(x^2 - 8x\), you take \(-8/2 = -4\), square it to get \(16\), then rewrite: \((x^2 - 8x + 16) - 16\).
- Factor the perfect square trinomial and simplify.
Coordinate geometry
In coordinate geometry, mathematically expressing shapes like parabolas involves plotting points on the x-y plane. The key concepts when graphing a quadratic equation include:
- The vertex, which is the point \((h, k)\) in vertex form \(y = a(x-h)^2 + k\).
- The reflection line, or axis of symmetry, is a vertical line that passes through the vertex, indicated by \(x = h\).
- Intercepts, where the graph crosses the axes. The y-intercept, for example, is found by setting \(x = 0\), and the x-intercepts (roots) can be found by solving the equation \(ax^2 + bx + c = 0\).
- The direction of opening, determined by the sign of \(a\): upward for positive \(a\) or downward for negative \(a\).
