Let's create the table of values and a graph for the given equation $y = 10 - x^2 - 3x$.

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1. Table of Values

We will calculate $y$ for a range of $x$-values. Here's the table for $x$ from $-5$ to $5$.

$x$	$y = 10 - x^2 - 3x$
$-5$	$y = 10 - (-5)^2 - 3(-5) = 10 - 25 + 15 = 0$
$-4$	$y = 10 - (-4)^2 - 3(-4) = 10 - 16 + 12 = 6$
$-3$	$y = 10 - (-3)^2 - 3(-3) = 10 - 9 + 9 = 10$
$-2$	$y = 10 - (-2)^2 - 3(-2) = 10 - 4 + 6 = 12$
$-1$	$y = 10 - (-1)^2 - 3(-1) = 10 - 1 + 3 = 12$
$0$	$y = 10 - (0)^2 - 3(0) = 10$
$1$	$y = 10 - (1)^2 - 3(1) = 10 - 1 - 3 = 6$
$2$	$y = 10 - (2)^2 - 3(2) = 10 - 4 - 6 = 0$
$3$	$y = 10 - (3)^2 - 3(3) = 10 - 9 - 9 = -8$
$4$	$y = 10 - (4)^2 - 3(4) = 10 - 16 - 12 = -18$

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2. Graph

The graph of the equation $y = 10 - x^2 - 3x$ is a downward-facing parabola because the coefficient of $x^2$ is negative. I’ll plot the graph based on the table. Let me generate this now. numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return 10 - x**2 - 3*x

Generate x values and calculate y values

x_values = np.linspace(-5, 5, 100) y_values = f(x_values)

Create the table data for specific points

x_table = np.array([-5, -4, -3, -2, -1, 0, 1, 2, 3, 4]) y_table = f(x_table)

Plot the graph

plt.figure(figsize=(8, 6)) plt.plot(x_values, y_values, label='y = 10 - x^2 - 3x', color='blue') plt.scatter(x_table, y_table, color='red', label='Table points') plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--') plt.title('Graph of y = 10 - x^2 - 3x') plt.xlabel('x') plt.ylabel('y') plt.legend() plt.grid(True) plt.show()Here is the graph of $y = 10 - x^2 - 3x$. The plotted red points correspond to the table values.

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Would you like more details on how the table was computed or about properties like the vertex and axis of symmetry?

Relative Questions:

1. How is the vertex of a parabola calculated, and what is the vertex here?
2. What role does the coefficient of $x^2$ play in the shape of the graph?
3. How can we find the roots of $y = 10 - x^2 - 3x$?
4. What are the applications of quadratic functions like this one?
5. How can we rewrite $y = 10 - x^2 - 3x$ in vertex form?

Tip:

To quickly identify key points of a parabola, focus on the vertex, the axis of symmetry, and intercepts (both x and y).