Question

How do you graph $f(x) = 3{x^2} + x - 5$?

Answer 1

Hint: According to the question given in the question we have to determine the graph for the$f(x) = 3{x^2} + x - 5$. So, to determine the graph for the given quadratic expression we have to substitute the different values of x which will be integers can be negative or positive to obtain the different values of y for the different values of x.
Now, we have to substitute the value of x as -6 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, we have to substitute the value of x as -4 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, we have to substitute the value of x as -2 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, we have to substitute the value of x as 0 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, we have to substitute the value of x as 2 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, we have to substitute the value of x as 4 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, we have to substitute the value of x as 6 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
Now, with the help of all the points obtained just above, we can determine the graph for the given expression.

Complete step by step solution:
Step 1: First of all we have to substitute the value of x as -6 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph. Hence,
$
   \Rightarrow y = 3{( - 6)^2} + ( - 6) - 5 \\
   \Rightarrow y = 108 - 11 \\
   \Rightarrow y = 97 \\
 $
Step 2: Now, we have to substitute the value of x as -4 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
$
   \Rightarrow y = 3{( - 4)^2} + ( - 4) - 5 \\
   \Rightarrow y = 48 - 9 \\
   \Rightarrow y = 39 \\
 $
Step 3: Now, we have to substitute the value of x as 0 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
$
   \Rightarrow y = 3{(0)^2} + (0) - 5 \\
   \Rightarrow y = - 5 \\
   \Rightarrow y = - 5 \\
 $
Step 4: Now, we have to substitute the value of x as 2 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
$
   \Rightarrow y = 3{(2)^2} + (2) - 5 \\
   \Rightarrow y = 12 - 3 \\
   \Rightarrow y = 9 \\
 $
Step 5: Now, we have to substitute the value of x as 4 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
$
   \Rightarrow y = 3{(4)^2} + (4) - 5 \\
   \Rightarrow y = 48 - 1 \\
   \Rightarrow y = 47 \\
 $
Step 6: Now, we have to substitute the value of x as 6 in the given function or we can say the quadratic equation to obtain the value of y so that we can achieve the points to plot in the graph.
$
   \Rightarrow y = 3{(6)^2} + (6) - 5 \\
   \Rightarrow y = 108 + 1 \\
   \Rightarrow y = 109 \\
 $
Step 7: Now, with the help of all the points we can draw a table so that we can easily draw the graph for the given expression. Hence,

x- value	$y = 3{x^2} + x - 5$
-6	97
-4	39
-2	5
0	-5
2	9
4	47
6	109

Now, with the help of the table above we can easily draw the graph for the given expression which is as drawn below.

Final solution: Hence, with the help of the points we have determined the graph for the given expression which is as below.

Note:
1. It is necessary that we have to substitute the different values of y as an integer which can be positive and negative both to determine the different values of y to obtain the points to plot in the graph.
2. To plot the point in the graph we have to take the points in the x-axis and y-axis on the quadrant plane so that we can draw the curve.