Problem 38
Question
Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=\frac{1}{2} x^{2}+3 x-7 $$
Step-by-Step Solution
Verified Answer
Thus, the graph of the function \(y = 0.5x^2 + 3x - 7\) opens upwards, the coordinates of the vertex are \(-3, -4.5\), and the equation of the axis of symmetry is \(x = -3\).
1Step 1: Determine the Opening of the Graph
For the given quadratic equation \(y = 0.5x^2 + 3x - 7\), the coefficient of \(x^2\) is 0.5, which is a positive number. Therefore, the graph of the given function opens upward.
2Step 2: Find the Coordinates of the Vertex
The coordinates of the vertex for any standard form quadratic equation can be found using the formula \((-b/2a , f(-b/2a))\). For the given equation \(y = 0.5x^2 + 3x - 7\), a = 0.5 and b = 3. Plugging this into the formula, we get the x-coordinate of the vertex as \(-3/(2*0.5) = -3\). To find the y-coordinate, substitute \(-3\) into the equation, getting \(y = 0.5(-3)^2 + 3(-3) - 7 = -4.5\). Hence, the coordinates of the vertex are \(-3, -4.5\).
3Step 3: Write an Equation of the Axis of Symmetry
The axis of symmetry for a standard form quadratic equation can be found using the formula \(x = -b/2a\). For the given equation \(y = 0.5x^2 + 3x - 7\), a = 0.5 and b = 3. Plugging this into the formula, we get the equation of the axis of symmetry as \(x = -3\).
Key Concepts
Vertex of a ParabolaAxis of SymmetryGraph Opens Up or Down
Vertex of a Parabola
The vertex of a parabola is an important point in the graph of a quadratic function. It represents the maximum or minimum value of the function, depending on the way the parabola opens.
For a quadratic equation in the standard form, which is \[ y = ax^2 + bx + c \]the vertex can be found using the formula \((-\frac{b}{2a}, f(-\frac{b}{2a}))\).
This formula gives us the x-coordinate first, calculated as \(-\frac{b}{2a}\), and then we use this value to find the corresponding y-coordinate.
In our example with \( y = 0.5x^2 + 3x - 7 \):
- The x-coordinate is calculated as \(-\frac{3}{2 * 0.5} = -3\).
- To find the y-coordinate, substitute \(-3\) for \(x\) in the equation, resulting in \(-4.5\).
Thus, the vertex of the parabola for this function is \((-3, -4.5)\), which is the lowest point since the parabola opens upward.
For a quadratic equation in the standard form, which is \[ y = ax^2 + bx + c \]the vertex can be found using the formula \((-\frac{b}{2a}, f(-\frac{b}{2a}))\).
This formula gives us the x-coordinate first, calculated as \(-\frac{b}{2a}\), and then we use this value to find the corresponding y-coordinate.
In our example with \( y = 0.5x^2 + 3x - 7 \):
- The x-coordinate is calculated as \(-\frac{3}{2 * 0.5} = -3\).
- To find the y-coordinate, substitute \(-3\) for \(x\) in the equation, resulting in \(-4.5\).
Thus, the vertex of the parabola for this function is \((-3, -4.5)\), which is the lowest point since the parabola opens upward.
Axis of Symmetry
The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. This line passes through the vertex, so it's crucial for understanding the parabola's shape.
For any parabola expressed in the standard quadratic form \( y = ax^2 + bx + c \), the axis of symmetry is always \( x = -\frac{b}{2a} \).
Using our example equation \( y = 0.5x^2 + 3x - 7 \):
- We compute the axis of symmetry as \( x = -\frac{3}{2 * 0.5} = -3 \).
This means the line of symmetry runs vertically through \(x = -3\).
The importance of the axis of symmetry is that it helps determine the balanced nature of the parabola on either side of this line.
For any parabola expressed in the standard quadratic form \( y = ax^2 + bx + c \), the axis of symmetry is always \( x = -\frac{b}{2a} \).
Using our example equation \( y = 0.5x^2 + 3x - 7 \):
- We compute the axis of symmetry as \( x = -\frac{3}{2 * 0.5} = -3 \).
This means the line of symmetry runs vertically through \(x = -3\).
The importance of the axis of symmetry is that it helps determine the balanced nature of the parabola on either side of this line.
Graph Opens Up or Down
Determining whether the parabola opens upwards or downwards depends on the sign of the coefficient of the \(x^2\) term in the quadratic function. This property defines the basic "U" shape of the parabola.
- If the coefficient \(a\) of \(x^2\) is positive, the parabola opens upwards like a smile.
- If \(a\) is negative, the parabola opens downwards like a frown.
In our given equation \( y = 0.5x^2 + 3x - 7 \), \( a = 0.5 \), which is positive.
This tells us that the parabola opens up, indicating that the vertex \((-3, -4.5)\) is the lowest point (or a minimum).
This behavior is critical for understanding the direction in which the parabola stretches on a graph.
- If the coefficient \(a\) of \(x^2\) is positive, the parabola opens upwards like a smile.
- If \(a\) is negative, the parabola opens downwards like a frown.
In our given equation \( y = 0.5x^2 + 3x - 7 \), \( a = 0.5 \), which is positive.
This tells us that the parabola opens up, indicating that the vertex \((-3, -4.5)\) is the lowest point (or a minimum).
This behavior is critical for understanding the direction in which the parabola stretches on a graph.
Other exercises in this chapter
Problem 38
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Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values. $$a=4, b=5, c=1$$
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