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Magazine Chapter 0: Problem 24
Graph. $$g(x)=-2 x^{2}-3 x+7$$
Short Answer
Expert verified
Vertex: (3/4, 3.625), y-intercept: (0, 7), opens downward.
Step by step solution
01
Identify the type of function
The given function is a quadratic function of the form \(g(x) = -2x^2 - 3x + 7\). Quadratic functions are represented graphically as parabolas.
02
Determine the direction of the parabola
Since the coefficient of \(x^2\) is negative (-2), the parabola opens downwards.
03
Find the vertex of the parabola
The vertex form of a quadratic function is given by \(g(x) = a(x-h)^2 + k\), where (h, k) is the vertex. For the standard form \(g(x) = ax^2 + bx + c\), the vertex can be found using \(h = -\frac{b}{2a}\) and \(k = g(h)\). Substituting \(a = -2\) and \(b = -3\), we get: \[ h = -\frac{-3}{2(-2)} = \frac{3}{4} \]. Substituting \(x = \frac{3}{4}\) into the equation gives us \(k\). Thus, \(k = -2 \left( \frac{3}{4} \right)^2 -3 \left( \frac{3}{4} \right) + 7\). Simplifying, \(k = -\frac{18}{16} - \frac{9}{4} + 7\) \(k = -\frac{9}{8} - \frac{9}{4} + 7\) \(k = -1.125 - 2.25 + 7 = 3.625\). Therefore, the vertex is \( \left( \frac{3}{4}, 3.625 \right) \).
04
Find the y-intercept
The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) in \(g(x)\), we get: \(g(0) = -2(0)^2 - 3(0) + 7 = 7\). Thus, the y-intercept is (0, 7).
05
Plot the symmetric points
Since the parabola is symmetric with respect to the vertex, we choose points around the vertex (\(x = \frac{3}{4}\)) and calculate their corresponding y-values. For example, choose \(x = 0\) and \(x = 1.5\): \(g(0) = 7\) and \(g(1.5) = -2(1.5)^2 - 3(1.5) + 7 = -4.5 - 4.5 + 7 = -2\). Plot these points: (0, 7) and (1.5, -2). Using symmetry, the point opposite 0 about \(x = \frac{3}{4}\) is\( x = -0.75\) and the point opposite 1.5 about \(x = \frac{3}{4}\) is -0.25.
06
Draw the parabola
Using the vertex, y-intercept, and symmetric points, sketch the parabola. Ensure it opens downward and passes through the plotted points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Function
A quadratic function is a type of polynomial function that is characterized by the squared term. It follows the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
The graph of a quadratic function is a parabola, which can either open upwards or downwards. The direction of the parabola is determined by the sign of the coefficient \(a\):
The graph of a quadratic function is a parabola, which can either open upwards or downwards. The direction of the parabola is determined by the sign of the coefficient \(a\):
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
Vertex Calculation
The vertex of a quadratic function represents the highest or lowest point on the graph, depending on the direction of the parabola.
For the given function \(g(x) = -2x^2 - 3x + 7\), the vertex can be found using the formula \(h = -\frac{b}{2a}\) for the x-coordinate and then substituting this value back into the original function to find the y-coordinate.
For this quadratic equation:
For the given function \(g(x) = -2x^2 - 3x + 7\), the vertex can be found using the formula \(h = -\frac{b}{2a}\) for the x-coordinate and then substituting this value back into the original function to find the y-coordinate.
For this quadratic equation:
- The coefficients are \(a = -2\) and \(b = -3\).
- The x-coordinate of the vertex \(h = -\frac{-3}{2 \times -2}=\frac{3}{4}\).
- Substitute \(x = \frac{3}{4}\) into the function to find the y-coordinate: \k = -2(\frac{3}{4})^2 - 3(\frac{3}{4}) + 7 \ k = -\frac{18}{16} - \frac{9}{4} + 7 \ k = 3.625.\
Parabola Direction
The direction in which the parabola opens is crucial for understanding the behavior of quadratic functions.
As mentioned earlier, the sign of the coefficient \(a\) determines this direction:
As mentioned earlier, the sign of the coefficient \(a\) determines this direction:
- If \(a\) is positive, the parabola opens upwards, indicating a minimum point at the vertex.
- If \(a\) is negative, the parabola opens downwards, indicating a maximum point at the vertex.
Y-Intercept
The y-intercept is the point where the graph of the function crosses the y-axis.
To find the y-intercept, set \(x = 0\) and solve for \(g(x)\). For the function \(g(x) = -2x^2 - 3x + 7\):
To find the y-intercept, set \(x = 0\) and solve for \(g(x)\). For the function \(g(x) = -2x^2 - 3x + 7\):
- Substitute \(x = 0\): \g(0) = -2(0)^2 - 3(0) + 7 \ g(0) = 7,\
Symmetric Points
The graph of a quadratic function (a parabola) is symmetric around its vertex. This means for every point \((x, y)\) on one side of the vertex, there is a corresponding point \((x\prime, y)\) on the other side where \(x - h = h - x\prime\), with \(h\) being the x-coordinate of the vertex.
Given the vertex \( (\frac{3}{4}, 3.625)\), if you choose points near the vertex like \( x = 0\) and \( x = 1.5\):
Given the vertex \( (\frac{3}{4}, 3.625)\), if you choose points near the vertex like \( x = 0\) and \( x = 1.5\):
- For \( x = 0\), \( g(0) = 7\)
- For \( x = 1.5\): \g(1.5) = -2(1.5)^2 - 3(1.5) + 7 \g(1.5) = -2. \
