Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=2 x-3\)

Step-by-Step Solution

Answer

The y-intercept is -3 and the x-intercept is 1.5. The graph of the given function is not symmetric. It's an increasing straight line passing through the points (0,-3) and (1.5,0).

Step 1: Figure Out the Intercepts

The y-intercept is the point where the line crosses the y-axis. This occurs when \(x = 0\). Substituting \(x = 0\) into the equation \(y = 2x - 3\) we get \(y = 2*0 - 3 = -3\). So the y-intercept is -3. The x-intercept is the point where the line crosses the x-axis. This occurs when \(y = 0\). Substituting \(y = 0\) into the same equation, we can solve for \(x = 1.5\). So the x-intercept is 1.5.

Step 2: Check for Symmetry

To check for symmetry in a linear equation, plug \(-x\) for \(x\) and see if the equation remains the same. Substituting \(-x\) for \(x\) in \(y = 2x - 3\), we get \(y = -2x - 3\), which is not equivalent to the original equation. Therefore, the given function is not symmetric.

Step 3: Sketch the Line

Using the intercepts, plot the two points on a graph, one point at (0,-3) and another at (1.5,0). Then draw a straight line passing through the two points. A straight line from bottom left to top right indicates that the function is increasing. Since the function is not symmetric, there is no axis of symmetry.

Key Concepts

Understanding the Y-Intercept

The y-intercept of a linear equation is a fundamental element that represents where the line crosses the y-axis. This is easily found when the value of x is zero. For the equation
\(y = 2x - 3\),
by setting \(x = 0\) we determine that the y-intercept is \(y = -3\). This means that the point (0, -3) is on the line and it is where the graph cuts through the vertical y-axis.

To visualize this, imagine plotting the graph on a coordinate plane: the y-intercept is where your pencil first touches the paper if you start drawing the line from the y-axis. This point is crucial for graphing because it serves as a starting position for the line.

Decoding the X-Intercept

On the flip side, the x-intercept indicates where the line meets the x-axis. This happens when the y-value is zero. From the same equation,
\(y = 2x - 3\),
by letting \(y = 0\) and solving for x, we find the x-intercept to be \(x = 1.5\). Consequently, the point where the line intercepts the x-axis is (1.5, 0).

Graphically, the x-intercept is the point where if the graph were a tightrope, it would touch the horizontal 'ground' of the x-axis. It's helpful for plotting because it provides another fixed point from which to draw your line accurately.

Exploring Symmetry in Equations

Symmetry in equations is a property that shows a balanced and mirror-like quality in the graph. For linear equations, this symmetry is checked by substituting \(-x\) for \(x\). If the result yields the original equation, the line is symmetric with respect to the y-axis. In our case, substituting \(-x\) in
\(y = 2x - 3\),
gives us \(y = -2x - 3\), which is not identical to the starting equation. Hence, this line isn't symmetrical, implying that folding the graph along the y-axis wouldn't produce a perfect overlap.

Symmetry is more commonly found in quadratic or higher-order polynomial functions, but knowing how to test for it in linear functions is a good skill for understanding complex graphs.

Graphing Lines with Precision

Graphing lines entails plotting two or more points and connecting them with a straight line. With our equation
\(y = 2x - 3\),
we've identified two crucial points: the y-intercept (0, -3) and the x-intercept (1.5, 0). To sketch the graph, start by plotting these intercepts on the coordinate plane. Next, use a ruler to draw a line passing through them, extending it to both ends of the graph.

Remember that the orientation of the line reveals the function's behavior: a line that slants upwards from left to right, like ours, means the function is increasing. It's a simple process, but accuracy is key. Every line tells a story, and these points are the main characters.