Problem 28 Graph each equation by plotting ... [FREE SOLUTION]

Chapter 1: Problem 28

Graph each equation by plotting points that satisfy the equation. $$3 x^{2}+2 y=-4$$

Short Answer

Expert verified

In order to graph the equation, transform the equation to express y in terms of x, select values for x, calculate the corresponding y values, and plot these points.

Step by step solution

Step 1:Rewrite Equation in Terms of Y

Since we need to plot points, it's easier to express y in terms of x. From the equation $3x^2 + 2y = -4$, now we can solve for y by subtracting $3x^2$ from both sides to get: $2y = -3x^2 - 4$. Finally divide by 2 to express y in terms of x : $y = -1.5x^2 - 2$.

Choose Values for X and Solve for Y

To create a sketch of the curve, select some values of x and solve for y. For example, choose x=-2, -1, 0, 1, 2. Calculate the corresponding values of y using the equation in Step 1.

Plot the Points on the Graph

Now, plot these points on the graph paper. Finally, draw a smooth curve that joins these points. The plot represents the graph of the given equation.

Validate

Make sure the plotted points follow the general shape of a quadratic function, which should be a parabola opening downward since the coefficient of $x^2$ is negative in the original equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Points

Plotting points is a fundamental method to draw graphs of functions such as quadratic equations. The idea is to find specific coordinate points

where each point corresponds to a value of $x$ and its resulting $y$.
By marking these points on the Cartesian plane, you can visualize the function.

To plot points for a quadratic equation like $y = -1.5x^2 - 2$:
1. Choose various values for $x$ (such as $-2, -1, 0, 1, 2$). 2. Substitute each $x$ value into the equation to compute the corresponding $y$.
Once you have the coordinates, place them accurately on the graph paper. Stitch these points into a smooth curve to depict the overall path of the equation. This method helps not just in sketching the graph, but also in understanding the behavior of the quadratic function as $x$ changes.

Quadratic Functions

Quadratic functions are equations of the form $ax^2 + bx + c = 0$. They are one of the simplest types of polynomial equations, with the highest degree term being $x^2$. The general equation includes three coefficients:

$a$, which affects the width and the direction of the opening of the parabola,
$b$, influencing the position horizontally
and $c$, which establishes the vertical shift of the parabola.

In the equation $y = -1.5x^2 - 2$, we observe:

A single variable: $x$,
And two constants: where $a = -1.5$ and $c = -2$.

The coefficient $-1.5$ indicates that the parabola opens downwards, as negative $a$ values always result in downward opening. Additionally, altering $c$ will shift the entire curve up or down the coordinate plane.
Quadratic functions always graph as parabolas, showcasing symmetric properties as they extend identically on both sides from their central vertex.

Parabola Characteristics

The parabola is the visual representation of a quadratic function on a graph. This results in several key characteristics to understand:

Vertex: The peak or lowest point of the parabola depending on the orientation.
Axis of Symmetry: A vertical line through the vertex, dividing the parabola into two mirror images.
For orientation: Direction of Opening: Indicated by the sign of the coefficient $a$ in the quadratic equation. A negative value suggests the parabola opens downwards.

The graph $y = -1.5x^2 - 2$ opens downward, meaning it forms a 'U' shape that falls as you move from zero outwards in any direction.
Another feature worth noting is the width of the parabola defined by the value of $a$. When $a$ is less than $1$, like $-1.5$, the curve appears to stretch a bit wider compare to standard parabolas.
These characteristics help in predicting the behavior of quadratic functions, including estimating where graphs might intercept the x-axis and y-axis.

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91影视

Graph each equation by plotting points that satisfy the equation. $$3 x^{2}+2 y=-4$$

Short Answer

Step by step solution

Step 1:Rewrite Equation in Terms of Y

Choose Values for X and Solve for Y

Plot the Points on the Graph

Validate

Key Concepts

Plotting Points

Quadratic Functions

Parabola Characteristics

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