Problem 4

Question

Exer. 1-20: Sketch the graph of the equation, and label the $x$ - and $y$-intercepts. $$ y=-2 x-3 $$

Step-by-Step Solution

Verified

Answer

Y-intercept: (0, -3); X-intercept: $-\frac{3}{2}, 0$.

1Step 1: Understand the Form of the Equation

The given equation is in the form of a linear equation, which is $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. Here, the equation is $ y = -2x - 3 $.

2Step 2: Identify the Y-intercept

The y-intercept is the value of $ y $ when $ x $ is 0. By setting $ x = 0 $ in the equation, we have $ y = -3 $. Thus, the y-intercept is $ (0, -3) $.

3Step 3: Find the X-intercept

The x-intercept is the value of $ x $ when $ y $ is 0. Set $ y = 0 $ in the equation and solve for $ x $:\[ 0 = -2x - 3 \]Add 3 to both sides:\[ 3 = -2x \]Divide by -2:\[ x = -\frac{3}{2} \]Thus, the x-intercept is $ \left(-\frac{3}{2}, 0\right) $.

4Step 4: Sketch the Graph

Plot the x-intercept $ \left(-\frac{3}{2}, 0\right) $ and the y-intercept $ (0, -3) $ on the coordinate plane. Draw a straight line through these two points since the equation represents a linear function.

5Step 5: Review the Graph

Make sure the line has a downward slope, consistent with the negative slope $ m = -2 $. The line passes through the intercept point $ (0, -3) $ and $ \left(-\frac{3}{2}, 0\right) $, confirming correctness.

Key Concepts

graphing linear equationsx-interceptsy-intercepts

graphing linear equations

Graphing a linear equation involves creating a visual representation of the equation on a coordinate plane. Linear equations are represented by straight lines, and they typically follow the format:

$ y = mx + b $

Here, $ m $ is the slope of the line, indicating how steep it is, while $ b $ is the y-intercept, the point where the line crosses the y-axis.To graph a linear equation such as $ y = -2x - 3 $, start by identifying key points like the x- and y-intercepts. Once these points are plotted, draw a straight line through them. This line visually represents all the possible solutions to the equation. Remember, since $ m = -2 $ is negative in this example, the slope of the line should be downward from left to right.

x-intercepts

X-intercepts are where the graph of the equation crosses the x-axis. At these points, the value of $ y $ is zero. To find the x-intercept for $ y = -2x - 3 $, replace $ y $ with 0 and solve for $ x $:

Solve $ 0 = -2x - 3 $, by adding 3 to both sides:\[ 3 = -2x \]Then divide by -2:\[ x = -\frac{3}{2} \]This calculation shows that the x-intercept is $ \left(-\frac{3}{2}, 0\right) $. Knowing how to find the x-intercept is crucial for graphing because it provides a critical point through which the line will pass.

y-intercepts

The y-intercept is the point at which the graph crosses the y-axis. At the y-intercept, the value of $ x $ is zero. To determine the y-intercept of $ y = -2x - 3 $, substitute $ x = 0 $ into the equation:Substitute as follows:\[ y = -2(0) - 3 \]\[ y = -3 \]This tells us that the y-intercept is $ (0, -3) $. Plotting the y-intercept on the graph is a great start to visualizing the linear equation. Like x-intercepts, y-intercepts help define the trajectory of the graphed line and are essential for accurately drawing linear relationships.

Problem 4

Other exercises in this chapter

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Problem 4

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Problem 4

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Problem 5

Exer. 3-12: Determine whether $f$ is even, odd, or neither even nor odd. $$ f(x)=3 x^{4}+2 x^{2}-5 $$

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