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Chapter 2: Problem 36

Graphical Analysis In Exercises \(35-42,\) use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). Then check your results algebraically by writing the quadratic function in standard form. $$ f(x)=-\left(x^{2}+x-30\right) $$

Short Answer

Expert verified
The vertex, x-intercepts and axis of symmetry are found by graphing the function and solving algebraically. The function can be written in standard form as \(f(x) = a(x - h)^2 + k\).

Step by step solution

01

Graph the quadratic function

Use a graphing utility to graph the quadratic function \( f(x)=-\left(x^{2}+x-30\right) \). The graph represents a downward-opening parabola because of the negative coefficient at \(x^2\).
02

Identify the vertex

The vertex of a parabola \( y = a(x - h)^2 + k \) is given by the point (h, k). Rewrite the given equation in this form by completing the square to find the vertex. The vertex will be the maximum point since the parabola opens downward.
03

Identify the x-intercepts

The x-intercepts of the function are points where the graph crosses the x-axis. These are found when \(f(x) = 0\). Solve \(f(x) = 0\) to find the x-intercepts.
04

Identify the axis of symmetry

The axis of symmetry is the vertical line that passes through the vertex of the parabola. Its equation is \(x = h\), where \(h\) is the x-coordinate of the vertex.
05

Rewrite the function in standard form

The standard form of a quadratic function is \(f(x) = a(x - h)^2 + k\), where (h, k) is the vertex. Complete the square on the given function \(f(x) = -\left(x^{2}+x-30\right)\) to rewrite it in standard form. This provides a check on the graphical results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Parabola
Understanding the vertex of a parabola is essential when graphing quadratic functions. The vertex is the highest or lowest point on the parabola, serving as a 'turning point' for the graph. For the equation in the form
\( y=a(x-h)^2+k \), the vertex is given by the point \((h, k)\). For the function \( f(x)=-\big(x^{2}+x-30\big) \), by completing the square, we can find the vertex to visually determine the graph鈥檚 maximum or minimum value. Since the parabola opens downwards (due to the negative coefficient of \( x^2 \)), the vertex represents the maximum point.
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror images, reflecting across this axis. For any parabola described by \( y = a(x - h)^2 + k \), the axis of symmetry is the line \( x = h \), with \( h \) being the x-coordinate of the vertex.
In our example, once we determine \( h \) by rewriting the function in vertex form, we can easily find the axis of symmetry. This helps identify the balanced structure of the parabola and predict the position of points on either side of the axis.
X-Intercepts of a Quadratic Function
The x-intercepts, also known as the zeros of the function, are the points where the parabola crosses the x-axis, which is found by solving \( f(x) = 0 \). For \( f(x) = -\big(x^{2}+x-30\big) \), we solve this equation to find the values of \( x \) where the graph meets the x-axis. These intercepts are critical for sketching the parabola's shape and understanding the function's real-world implications, such as breaking even in a business model.
Standard Form of a Quadratic Function
The standard form of a quadratic function is an expression \( f(x) = a(x - h)^2 + k \), revealing the vertex \( (h, k) \) and the direction in which the parabola opens (upward if \( a > 0 \), downward if \( a < 0 \)). It provides a clear representation to identify key features of the parabola. Converting a function like \( f(x) = -\big(x^{2}+x-30\big) \) to standard form not only helps in graphing it but also simplifies the process of analyzing its properties by exposing the vertex directly.
Completing the Square
Completing the square is a method used to rewrite a quadratic function in standard form. By transforming \( ax^2 + bx + c \) into \( a(x - h)^2 + k \), we gain valuable insights into the parabola's vertex and axis of symmetry.
For our function \( f(x)=-\big(x^{2}+x-30\big) \), we鈥檇 add and subtract a certain number inside the parenthesis to create a perfect square trinomial, facilitating the equation鈥檚 simplification and revealing the vertex form. This algebraic technique is pivotal for graph analysis and for solving quadratic equations.

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