Problem 18
Question
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ h(t)=-4 t^{2}+6 t-1 $$
Step-by-Step Solution
Verified Answer
Maximum value is \( \frac{3}{4} \) at \( t = \frac{3}{4} \). Axis of symmetry is \( t = \frac{3}{4} \).
1Step 1: Identify the Form of the Quadratic Function
The given function is a quadratic function in the form of \( h(t) = at^2 + bt + c \), where \( a = -4 \), \( b = 6 \), and \( c = -1 \).
2Step 2: Determine the Direction of the Parabola
Since the coefficient \( a = -4 \) is negative, the parabola opens downwards. This means the quadratic function has a maximum value.
3Step 3: Find the Axis of Symmetry
The axis of symmetry for a quadratic function \( at^2 + bt + c \) is given by \( t = -\frac{b}{2a} \). Here, \( b = 6 \) and \( a = -4 \), so the axis is \( t = -\frac{6}{2(-4)} = \frac{6}{8} = \frac{3}{4} \).
4Step 4: Calculate the Maximum Value
Substitute \( t = \frac{3}{4} \) back into the function to find the maximum value. \[ h\left(\frac{3}{4}\right) = -4\left(\frac{3}{4}\right)^2 + 6\left(\frac{3}{4}\right) - 1 \] Simplify this to get \[ h\left(\frac{3}{4}\right) = -\frac{9}{4} + \frac{18}{4} - 1 \] which further simplifies to \( \frac{3}{4} \).
5Step 5: Summarize Results
The parabola has a maximum value of \( \frac{3}{4} \) at \( t = \frac{3}{4} \) with an axis of symmetry at \( t = \frac{3}{4} \).
Key Concepts
Maximum Value in a Quadratic FunctionUnderstanding the Axis of SymmetryDirection of the Parabola
Maximum Value in a Quadratic Function
In a quadratic function, determining whether there is a minimum or maximum value is crucial for graph interpretation. Quadratic functions typically take the form \( ax^2 + bx + c \). A key player in figuring out whether we have a maximum or minimum on our hands is the coefficient \( a \).
A negative \( a \), as in the quadratic function \( h(t) = -4t^2 + 6t - 1 \), indicates that the parabola will open downwards, forming a hill rather than a valley. This means it possesses a **maximum value** at its peak.
To pinpoint this maximum value, we first calculate the vertex of the parabola. The vertex lies at the axis of symmetry, where the maximum or minimum value of the quadratic is found. In the case of our function, once we determine the axis of symmetry which occurs at \( t = \frac{3}{4} \), we substitute this value back into the quadratic equation to find \( h(\frac{3}{4}) \). We then find that the maximum value is \( \frac{3}{4} \). This is where the function reaches its peak before its values decrease on either side.
Therefore, whenever you're dealing with a quadratic function with a negative \( a \), remember: you're heading for a max!
A negative \( a \), as in the quadratic function \( h(t) = -4t^2 + 6t - 1 \), indicates that the parabola will open downwards, forming a hill rather than a valley. This means it possesses a **maximum value** at its peak.
To pinpoint this maximum value, we first calculate the vertex of the parabola. The vertex lies at the axis of symmetry, where the maximum or minimum value of the quadratic is found. In the case of our function, once we determine the axis of symmetry which occurs at \( t = \frac{3}{4} \), we substitute this value back into the quadratic equation to find \( h(\frac{3}{4}) \). We then find that the maximum value is \( \frac{3}{4} \). This is where the function reaches its peak before its values decrease on either side.
Therefore, whenever you're dealing with a quadratic function with a negative \( a \), remember: you're heading for a max!
Understanding the Axis of Symmetry
The axis of symmetry is a magical line in quadratic functions that evenly splits the parabola into two mirror-image halves. It's like the spine of the parabola, guiding us straight to the vertex. Think of it as the fulcrum of a seesaw that ensures balance.
The axis of symmetry for any quadratic function \( ax^2 + bx + c \) can be calculated with a simple formula: \( x = -\frac{b}{2a} \).
In the function \( h(t) = -4t^2 + 6t - 1 \), we plug in \( b = 6 \) and \( a = -4 \), which gives us the axis of symmetry \( t = \frac{3}{4} \). This line doesn’t just divide the parabola; it also represents the midpoint along the x-axis where the vertex - the highest or lowest point of the function - is located.
Understanding the axis of symmetry helps you easily find the maximum or minimum point and is an indispensable tool when sketching the graph of a quadratic function.
The axis of symmetry for any quadratic function \( ax^2 + bx + c \) can be calculated with a simple formula: \( x = -\frac{b}{2a} \).
In the function \( h(t) = -4t^2 + 6t - 1 \), we plug in \( b = 6 \) and \( a = -4 \), which gives us the axis of symmetry \( t = \frac{3}{4} \). This line doesn’t just divide the parabola; it also represents the midpoint along the x-axis where the vertex - the highest or lowest point of the function - is located.
Understanding the axis of symmetry helps you easily find the maximum or minimum point and is an indispensable tool when sketching the graph of a quadratic function.
Direction of the Parabola
The direction a parabola opens is determined by the sign of the coefficient \( a \) in the quadratic expression \( ax^2 + bx + c \). This tiny detail tells us whether we’re looking at a smile (opens upwards) or a frown (opens downwards) on a graph.
For the quadratic function \( h(t) = -4t^2 + 6t - 1 \), our friend \( a \) equals \(-4\), which is negative. Hence, the parabola opens **downwards**. This downward-opening frown implies the graph resembles an arch, extending outward into infinity while culminating at a high point, known as a maximum.
If that \( a \) value were positive, however, the situation would flip: the parabola would open upwards, resembling a bowl or smile, with the lowest point acting as the minimum. Recognizing this direction upfront helps shape a clear image of what the graph looks like and where its peak or dip is positioned.
For the quadratic function \( h(t) = -4t^2 + 6t - 1 \), our friend \( a \) equals \(-4\), which is negative. Hence, the parabola opens **downwards**. This downward-opening frown implies the graph resembles an arch, extending outward into infinity while culminating at a high point, known as a maximum.
If that \( a \) value were positive, however, the situation would flip: the parabola would open upwards, resembling a bowl or smile, with the lowest point acting as the minimum. Recognizing this direction upfront helps shape a clear image of what the graph looks like and where its peak or dip is positioned.
- Positive \( a \): Parabola opens upwards
- Negative \( a \): Parabola opens downwards
Other exercises in this chapter
Problem 18
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