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Problem 551

# Graph the function $$3 \mathrm{x}^{2}+5 \mathrm{x}-7$$

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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