Chapter 19: Chapter 19

Problem 551

Graph the function \(3 \mathrm{x}^{2}+5 \mathrm{x}-7\)

Short Answer

Expert verified

The vertex of the quadratic function \(3x^2 + 5x - 7\) is \(\left(\frac{-5}{6}, -3.08\right)\), with an axis of symmetry at \(x = \frac{-5}{6}\), and the parabola opens upwards. Plot the vertex, axis of symmetry, and some additional points obtained from the function to sketch the graph of the function accurately.

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Step 1: Determine the vertex

To determine the vertex of the quadratic function given in the form \(f(x) = ax^2 + bx + c\), we'll use the formula for the vertex coordinates \((h,k)\): \[h = \frac{-b}{2a}\] \[k = f(h)\] In our case, \(a = 3\), \(b = 5\), and \(c = -7\). Plugging the values of a and b into the formula to find h: \[h = \frac{-5}{2(3)} = \frac{-5}{6}\] Now, we will find the y-coordinate (k) by plugging h back into the function: \[k = f\left(\frac{-5}{6}\right) = 3\left(\frac{-5}{6}\right)^2 + 5\left(\frac{-5}{6}\right) - 7\] Calculating k: \[k ≈ -3.08\] So, the vertex of the function is at the point \(\left(\frac{-5}{6}, -3.08\right)\).

Step 2: Identify the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. Its equation is given by \(x = h\), where h is the x-coordinate of the vertex. In our case, the axis of symmetry is given by: \[x = \frac{-5}{6}\]

Step 3: Determine the direction of the parabola

The direction of the parabola is determined by the coefficient a of the quadratic function. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), the parabola opens downwards. In our case, since \(a = 3\) is positive, the parabola opens upwards.

Step 4: Plot the vertex, axis of symmetry, and other points

Now that we have the vertex, axis of symmetry, and direction of the parabola, we can plot these features on a graph: 1. Plot the vertex point \(\left(\frac{-5}{6}, -3.08\right)\) on the graph. 2. Draw the axis of symmetry as a dashed vertical line at \(x = \frac{-5}{6}\). 3. Notice that the parabola opens upwards. To make an accurate sketch of the graph, we need to find a few additional points: 4. Plug in a few x-values close to the vertex (both on the left and on the right) into the function to find corresponding y-values, and then plot these points on the graph. 5. Reflect the points you plotted in step 4 across the axis of symmetry to find additional points on the parabola. 6. Draw the parabola by connecting the plotted points with a smooth curve. With all these key features and points plotted on the graph, you have successfully graphed the function \(3x^2 + 5x - 7\).

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Chapter 19: Chapter 19

Graph the function \(3 \mathrm{x}^{2}+5 \mathrm{x}-7\)

Short Answer

Step by step solution

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Step 1: Determine the vertex

Step 2: Identify the axis of symmetry

Step 3: Determine the direction of the parabola

Step 4: Plot the vertex, axis of symmetry, and other points

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