Open in App
Log In Start studying!

Select your language

Suggested languages for you:

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.
See the step by step solution

Step by step solution

Unlock all solutions

Get unlimited access to millions of textbook solutions with Vaia Premium

Over 22 million students worldwide already upgrade their learning with Vaia!

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 \(3x^2 + 5x - 7\).

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Access millions of textbook solutions in one place

  • Access over 3 million high quality textbook solutions
  • Access our popular flashcard, quiz, mock-exam and notes features
  • Access our smart AI features to upgrade your learning
Get Vaia Premium now
Access millions of textbook solutions in one place

Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

  • Flashcards & Quizzes
  • AI Study Assistant
  • Smart Note-Taking
  • Mock-Exams
  • Study Planner
Join over 22 million students in learning with our Vaia App Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks