Problem 20
Question
Graph each of the following linear and quadratic functions. $$f(x)=-4 x^{2}-8 x$$
Step-by-Step Solution
Verified Answer
The graph is a downward-facing parabola with vertex (1, -12) and y-intercept (0, 0).
1Step 1: Identify the type of function
The given function is a quadratic function because it is defined as a polynomial with the highest degree of 2, i.e., \( f(x) = -4x^2 - 8x \). Quadratic functions generally have a parabolic shape.
2Step 2: Determine the standard form
The quadratic function \( f(x) = -4x^2 - 8x \) is already in the standard form \( ax^2 + bx + c \), where \( a = -4 \), \( b = -8 \), and \( c = 0 \). This allows us to proceed in finding important characteristics of its graph.
3Step 3: Find the vertex
The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Plugging in our values, we have:\[ x = -\frac{-8}{2(-4)} = 1 \]Substitute \( x = 1 \) back into the function to find \( y \):\[ f(1) = -4(1)^2 - 8(1) = -4 - 8 = -12 \]Thus, the vertex of the parabola is \((1, -12)\).
4Step 4: Determine the axis of symmetry
The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. For our function, it can be expressed as \(x = 1\). This axis divides the parabola into two symmetrical parts.
5Step 5: Find the y-intercept
The y-intercept of a function is the point where the graph crosses the y-axis, which occurs when \( x = 0 \). Calculate as follows:\[ f(0) = -4(0)^2 - 8(0) = 0 \]So, the y-intercept is at the point \((0, 0)\).
6Step 6: Sketch the graph
To sketch the graph, start by plotting the vertex \((1, -12)\) and the y-intercept \((0, 0)\). The parabola opens downwards because the leading coefficient \(a = -4\) is negative.Plot additional points either side of the axis and use the vertex and y-intercept to guide the parabolic shape. Ensure symmetry about the axis \(x = 1\).
7Step 7: Identify the shape and orientation
Since the coefficient \(a\) of \(x^2\) is negative, the parabola is downward facing. We confirmed the downward opening through additional calculated points fitting the form \(f(x) = -4x^2 - 8x\). By analyzing its symmetrical characteristics, full graph representation is possible.
Key Concepts
Vertex of a ParabolaAxis of SymmetryY-interceptParabola Orientation
Vertex of a Parabola
The vertex of a parabola is a key point that defines its shape and position on a graph. In a quadratic function of the form \( ax^2 + bx + c \), the vertex can be found using a specific formula. This point represents the maximum or minimum value of the function, depending on its orientation. To find the vertex, use the formula \( x = -\frac{b}{2a} \).
For the quadratic function given as \( f(x) = -4x^2 - 8x \), we substitute the coefficients \( a = -4 \) and \( b = -8 \) into the formula. This results in \( x = 1 \).
We then substitute \( x = 1 \) back into the function to determine the y-coordinate, resulting in \( f(1) = -12 \). Hence, the vertex of this parabola is \( (1, -12) \). The vertex marks either the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
For the quadratic function given as \( f(x) = -4x^2 - 8x \), we substitute the coefficients \( a = -4 \) and \( b = -8 \) into the formula. This results in \( x = 1 \).
We then substitute \( x = 1 \) back into the function to determine the y-coordinate, resulting in \( f(1) = -12 \). Hence, the vertex of this parabola is \( (1, -12) \). The vertex marks either the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
Axis of Symmetry
The axis of symmetry is an imaginary vertical line that runs through the vertex of a parabola. This line acts as a mirror, making sure that each side of the parabola is a mirror image of the other. It's a crucial concept in understanding the parabola's symmetrical nature.
For our quadratic function \( f(x) = -4x^2 - 8x \), we already calculated the vertex as \( (1, -12) \). Therefore, the axis of symmetry for this function is the vertical line \( x = 1 \).
Recognizing this line helps us anticipate the shape and direction of the parabola, making it easier to sketch and analyze. It's part of what makes the graphing of quadratic functions both interesting and manageable.
For our quadratic function \( f(x) = -4x^2 - 8x \), we already calculated the vertex as \( (1, -12) \). Therefore, the axis of symmetry for this function is the vertical line \( x = 1 \).
Recognizing this line helps us anticipate the shape and direction of the parabola, making it easier to sketch and analyze. It's part of what makes the graphing of quadratic functions both interesting and manageable.
Y-intercept
The y-intercept of a quadratic function is the point where the graph intersects the y-axis. This happens when the value of \( x \) is zero. To find the y-intercept, substitute \( x = 0 \) into the quadratic function.
For our function \( f(x) = -4x^2 - 8x \), when \( x = 0 \), the function simplifies to \( f(0) = 0 \).
Thus, the y-intercept is the point \( (0, 0) \), meaning the parabola passes through the origin. The y-intercept is a straightforward but essential aspect of graphing quadratics, making it easier to connect the graph to real-world applications.
For our function \( f(x) = -4x^2 - 8x \), when \( x = 0 \), the function simplifies to \( f(0) = 0 \).
Thus, the y-intercept is the point \( (0, 0) \), meaning the parabola passes through the origin. The y-intercept is a straightforward but essential aspect of graphing quadratics, making it easier to connect the graph to real-world applications.
Parabola Orientation
Orientation refers to the direction in which the parabola opens, either upwards or downwards. This is determined by the sign of the coefficient \( a \) in the quadratic function \( ax^2 + bx + c \).
If \( a \) is positive, the parabola opens upwards, resembling a 'U' shape. Conversely, if \( a \) is negative, it opens downwards like an upside-down 'U'. For our equation \( f(x) = -4x^2 - 8x \), \( a = -4 \), which confirms that the parabola opens downwards.
Knowing the orientation is crucial when sketching the graph. It immediately tells us whether to look for a maximum (for downward opening) or a minimum (for upward opening) point at the vertex. This characteristic influences everything from the graph's shape to the nature of solutions for the equation.
If \( a \) is positive, the parabola opens upwards, resembling a 'U' shape. Conversely, if \( a \) is negative, it opens downwards like an upside-down 'U'. For our equation \( f(x) = -4x^2 - 8x \), \( a = -4 \), which confirms that the parabola opens downwards.
Knowing the orientation is crucial when sketching the graph. It immediately tells us whether to look for a maximum (for downward opening) or a minimum (for upward opening) point at the vertex. This characteristic influences everything from the graph's shape to the nature of solutions for the equation.
Other exercises in this chapter
Problem 20
Show that \((f \circ g)(x)=x\) and \((g \circ f)\) \((x)=x\) for each pair of functions. \(f(x)=-2 x\) and \(g(x)=-\frac{1}{2} x\)
View solution Problem 20
Graph each of the functions. $$f(x)=\frac{1}{4} x^{2}$$
View solution Problem 20
Specify the domain for each of the functions. $$f(x)=\frac{7}{x^{2}-8 x-20}$$
View solution Problem 21
Find the constant of variation for each of the stated conditions. \(y\) varies directly as \(x\) and inversely as \(z\), and \(y=45\) when \(x=18\) and \(z=2\).
View solution