Find The Coordinates Of The Vertex For The Parabola

find The Coordinates Of The Vertex For The Parabola Defined And Give

The task involves analyzing quadratic functions, determining their key features, and applying various algebraic and graphing methods. Specifically, it includes finding the coordinates of the vertex, domain, and range in set-builder notation, understanding the parabola's axis of symmetry, sketching the graph using vertex and intercepts, analyzing the function's maximum or minimum value without graphing, and solving related quadratic equations through various methods such as factoring, completing the square, and the square root property. The assignment also requires interpreting the geometric and algebraic properties of parabolas, including their orientation, discriminant, and symmetry, as well as applying real-world models like projectile motion and optimization problems involving fencing and areas. Additionally, it involves determining when quadratic functions are even, odd, or neither and analyzing the symmetry of their graphs, along with solving quadratic equations algebraically and using their properties to find solutions.

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Quadratic functions are fundamental in mathematics, exhibiting characteristic parabolic shapes determined by their coefficients and vertex points. Understanding their features such as the vertex, axis of symmetry, domain, and range are essential for graphing and analyzing these functions, both algebraically and visually.

Finding the Vertex of a Parabola

The vertex of a parabola given by the quadratic function \(f(x) = a(x-h)^2 + k\) is located at the point \((h,k)\). Alternatively, for the standard form \(ax^2 + bx + c\), the vertex's x-coordinate is found using the formula \(-\frac{b}{2a}\). Substituting this value back into the function yields the y-coordinate.

For instance, consider \(f(x) = -3(x+2)^2 + 12\). Here, the vertex is at \((-2, 12)\), because the transformation indicates a horizontal shift of \(-2\) and a vertical shift up to 12.

Domain and Range in Set-Builder Notation

The domain of any quadratic function is all real numbers, expressed as \(\{x \mid x \in \mathbb{R}\}\). The range depends on whether the parabola opens upward or downward.

For example, in the case of \(f(x) = -3(x+2)^2 + 12\), since it opens downward (as the coefficient \(a=-3

Graphing Using Vertex and Intercepts

To graph a quadratic function effectively, plot the vertex and intercepts, then draw a smooth parabola through these points. Consider the intercepts by setting \(x=0\) to find the y-intercept, and the y-value to zero to find x-intercepts. The axis of symmetry passes through the vertex, expressed as \(x = h\).

For example, with \(g(x) = 2x^2 -7x - 4\), the vertex can be found using \(x = -\frac{b}{2a} = -\frac{-7}{2(2)} = \frac{7}{4}\). Substituting back gives the y-coordinate, and the intercepts are obtained by solving \(g(x) = 0\).

Determining the Maximum or Minimum Value

The parabola's concavity determines whether it has a maximum or minimum. If \(a > 0\), the parabola opens upward, and the vertex is a minimum point. If \(a

For example, \(h(x) = -2x^2 - 12x + 3\) opens downward (\(a = -2

Using the Vertex to Describe Domain and Range

Given a vertex \((-3, -4)\) and that the parabola opens downward, the domain remains all real numbers, \(\{x \mid x \in \mathbb{R}\}\). The range includes all y-values less than or equal to \(-4\), expressed as \(\{ y \mid y \leq -4 \}\).

Constructing a Quadratic Equation from Vertex

To create an equation with a given vertex, use the vertex form \(f(x) = a(x - h)^2 + k\). For example, if the vertex is \((-8, -6)\), and the parabola has the same shape as \(f(x) = 2x^2\), then the equation becomes \(f(x) = 2(x + 8)^2 - 6\), where \(a=2\). This reflects the parabola shifted horizontally and vertically from the parent shape.

Analyzing Symmetry — Even, Odd, or Neither

A function is even if \(f(-x) = f(x)\), indicating symmetry about the y-axis. It is odd if \(f(-x) = -f(x)\), indicating symmetry about the origin. Otherwise, it is neither.

Consider \(f(\alpha) = \alpha^4 - 2\alpha^2 + 1\). Since \(\alpha^4\) and \(\alpha^2\) are even functions, \(f(\alpha)\) is even, and the graph is symmetrical about the y-axis.

Modeling Projectile Motion with Parabolas

The height of a shot-put at various distances can be modeled with quadratic functions. For instance, \(g(x) = -0.04x^2 + 2.1x + 6\) describes the height in feet. The maximum height occurs at the vertex of the parabola, found via \(-\frac{b}{2a}\) in the quadratic.

Calculations show the maximum height, the horizontal distance where it occurs, and the total distance traveled, considering initial vertical and horizontal velocities. These models are essential in sports science for optimizing performance.

Optimization Problems — Fencing and Area

Given 200 feet of fencing bordering a river (no fencing on the river side), the goal is to maximize the enclosed area. If \(w\) is the width perpendicular to the river and \(l\) is the length parallel to the river, then the total fencing used is \(2w + l = 200\).

Express the area as \(A = w \times l\), then rewrite \(l = 200 - 2w\), so \(A(w) = w(200 - 2w) = 200w - 2w^2\). The maximum occurs at the vertex of this quadratic, at \(w = -\frac{b}{2a} = -\frac{200}{2 \times -2} = 50\). Hence, the optimal dimensions are \(w=50\) feet and \(l=200 - 2(50) = 100\) feet, yielding the largest area of \(A = 50 \times 100 = 5000\) square feet.

Verifying Functions and Intercepts

Support for the functions' properties involves analyzing their equations, including intercepts, orientation, discriminant, vertex, and axis of symmetry. For example, the quadratic \(f(x) = 4.5 - (x - 2)^2\) has a y-intercept at when \(x=0\), and x-intercepts where the quadratic equals zero.

Solving Quadratic Equations

Various methods such as factoring, completing the square, and using the square root property are essential for solving quadratic equations without a calculator. For example, solving \(2x^2 - 7 = 0\) involves isolating \(x^2 = \frac{7}{2}\), then taking square roots to find solutions. Completing the square involves rewriting the quadratic in perfect square form and solving for \(x\).

The quadratic discriminant, given by \(b^2 - 4ac\), indicates the nature of solutions: positive discriminant implies two real solutions, zero indicates one real solution, and negative implies no real solutions.

Application in Real-World Problems

The ladder height problem demonstrates the use of quadratics in collinearity and geometric constraints. Suppose a ladder of 57 feet reaches the roof, with its base 17 feet from the building; then the height of the building can be calculated by formulating a quadratic based on the Pythagorean theorem.

In the problem involving squares, increasing side length by 2 inches results in a new square with an area of 36 square inches. From this, the original side length is deduced, illustrating reverse-engineering the dimensions from area.

Conclusion

Quadratic functions are versatile tools that offer insights into geometric shapes, physical phenomena, and optimization problems. Mastery of the algebraic techniques such as vertex form, factoring, completing the square, and analyzing discriminants enables precise modeling and problem-solving. Their symmetry, domain, range, and maximum or minimum values reveal critical properties that are essential across mathematics, physics, engineering, and applied sciences. Understanding these concepts deepens comprehension and improves problem-solving capabilities in diverse contexts.

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