Problem 36 (a) find the vertex, the axis of... [FREE SOLUTION]

Chapter 11: Problem 36

(a) find the vertex, the axis of symmetry, and the maximum or minimum function value and (b) graph the function. $$ g(x)=2 x^{2}+5 x-1 $$

Short Answer

Expert verified

Vertex: $ -\frac{5}{4}, -\frac{33}{8} $; Axis of symmetry: $ x = -\frac{5}{4} $; Minimum value: $ -\frac{33}{8} $; Graph the function as an upward opening parabola.

Step by step solution

- Find the Vertex

To find the vertex of the parabola given by the function $g(x) = 2x^2 + 5x - 1$, use the vertex formula $x = -\frac{b}{2a}$. Here, $a = 2$ and $b = 5$. Plug these values into the formula: \[ x = -\frac{5}{2(2)} = -\frac{5}{4} \] Next, substitute $x = -\frac{5}{4}$ back into the function to find the y-coordinate of the vertex: \[ g\left(-\frac{5}{4}\right) = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) - 1 \] Calculate this to find the y-coordinate.

- Complete Vertex Calculation

First, compute $\left(-\frac{5}{4}\right)^2$: \[\left(-\frac{5}{4}\right)^2 = \frac{25}{16} \] Now substitute back: \[ g\left(-\frac{5}{4}\right) = 2 \cdot \frac{25}{16} + 5 \left(-\frac{5}{4}\right) - 1 = \frac{50}{16} - \frac{25}{4} - 1 \] Simplify the expression to get the exact y-coordinate of the vertex.

- Simplify and Write Vertex

Continue simplifying: \[ g\left(-\frac{5}{4}\right) = \frac{50}{16} - \frac{100}{16} - \frac{16}{16} = \frac{50 - 100 - 16}{16} = \frac{-66}{16} = -\frac{33}{8} \] Thus, the vertex of the parabola is at: \[ \left( -\frac{5}{4}, -\frac{33}{8} \right) \]

- Find the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the x-coordinate of the vertex. For $g(x) = 2x^2 + 5x - 1$, the axis of symmetry is \[ x = -\frac{5}{4} \]

- Determine Maximum/Minimum Value

Given that the parabola opens upwards (since the coefficient of $x^2$ in the function, which is $2$, is positive), the vertex represents the minimum point. Hence, the minimum value of the function is $ g\left(-\frac{5}{4}\right) = -\frac{33}{8} $

- Graph the Function

To graph the function $g(x) = 2x^2 + 5x - 1$, plot the vertex $\left( -\frac{5}{4}, -\frac{33}{8} \right)$. Draw the axis of symmetry at $x = -\frac{5}{4}$. Additionally, find the y-intercept by evaluating $g(0) = -1$ and plot additional points symmetrically around the vertex.

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Key Concepts

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

vertex formula

The vertex of a quadratic function plays a significant role in understanding its behavior. The vertex formula helps us find the coordinates of the vertex of a parabola given by a quadratic function of the form $ax^2 + bx + c$. It is given by $x = -\frac{b}{2a}$. Here, $a$ and $b$ are coefficients from the quadratic equation.

axis of symmetry

The axis of symmetry is a crucial concept in graphing quadratic functions. It refers to the vertical line that passes through the vertex of the parabola. This line divides the parabola into two mirror-image halves. For the function $g(x) = 2x^2 + 5x - 1$, the x-coordinate of the vertex is $-\frac{b}{2a}$, which gives us $-\frac{5}{4}$. Therefore, the equation of the axis of symmetry is $x = -\frac{5}{4}$. This line helps in verifying the parabola鈥檚 symmetry when plotting points.

minimum value

Every quadratic function has either a minimum or a maximum value, determined by the direction in which the parabola opens. If the coefficient of the $x^2$ term ($a$) is positive, the parabola opens upwards, and the vertex represents the minimum value. For our function $g(x) = 2x^2 + 5x - 1$, since $a = 2$ is positive, the vertex $ \left( -\frac{5}{4}, -\frac{33}{8} \right) $ gives the minimum value. Therefore, the minimum value of the function is $ g\left(-\frac{5}{4}\right) = -\frac{33}{8} $.

graphing quadratic functions

Graphing a quadratic function involves a few steps. Start with plotting the vertex, as it is the lowest or highest point of the graph depending on whether the parabola opens upwards or downwards. For $g(x) = 2x^2 + 5x - 1$, the vertex is at $\left( -\frac{5}{4}, -\frac{33}{8} \right)$. Next, draw the axis of symmetry at $x = -\frac{5}{4}$. Additionally, plot the y-intercept, which is the point where the graph crosses the y-axis. This can be found by evaluating $g(0) = -1$. Finally, choose additional points on either side of the axis of symmetry and plot them to get a clearer shape of the parabola. Connect all these points with a smooth curve to complete the graph.

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91影视

(a) find the vertex, the axis of symmetry, and the maximum or minimum function value and (b) graph the function. $$ g(x)=2 x^{2}+5 x-1 $$

Short Answer

Step by step solution

- Find the Vertex

- Complete Vertex Calculation

- Simplify and Write Vertex

- Find the Axis of Symmetry

- Determine Maximum/Minimum Value

- Graph the Function

Key Concepts

vertex formula

axis of symmetry

minimum value

graphing quadratic functions

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