Example Question - plotting points
Here are examples of questions we've helped users solve.
Graphing a Linear Equation on a Coordinate Plane
<p>Para resolver esta pregunta, necesitamos graficar la ecuación lineal \(y = 3x - 2\) en el plano de coordenadas. Primero, utilizando la tabla de valores, podemos ver algunos puntos que ya se han calculado y que se pueden trazar en la gráfica.</p> <p>Paso 1: Trazar los puntos (3, 7), (2, 4) y (-2, -8) en el plano de coordenadas. Cada punto corresponde a un valor de 'x' y el valor de 'y' resultante tras aplicar la ecuación \(y = 3x - 2\).</p> <p>Paso 2: Dibujar una línea recta que pase por estos puntos, ya que representan la solución a la ecuación lineal y cualquier punto en esta línea satisfará la ecuación \(y = 3x - 2\).</p> <p>El punto donde la línea cruza el eje 'y' es el intercepto en y, que para esta ecuación es -2, y esto indica que cuando \(x=0\), \(y=-2\).</p> <p>La pendiente de la línea es 3, indicando que por cada aumento en 1 en la dirección de 'x', 'y' aumentará en 3 unidades.</p>
Graphing a Line from Point-Slope Form Equation
To graph the line given by the equation \( y - 2 = \frac{-2}{5}(x - 4) \), you can follow these steps: 1. Recognize the equation form: The equation is in point-slope form, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope of the line. 2. Identify the slope and a point: From the given equation, the slope (\( m \)) is \( -\frac{2}{5} \) and the point (\( x_1, y_1 \)) is (4, 2). 3. Plot the given point: Start by plotting the point (4, 2) on the coordinate plane. This is a point through which the line will pass. 4. Use the slope to find another point: The slope \( -\frac{2}{5} \) means that for every 5 units you move to the right (in the positive x-direction), you move 2 units down (in the negative y-direction) because the slope is negative. You could also move 5 units to the left (negative x-direction) and 2 units up (positive y-direction) to find another point. 5. Plot the second point: For example, starting at point (4, 2) and moving 5 units to the right, you reach the point (9, 0). Plot this point on the coordinate plane. 6. Draw the line: Once you have these two points plotted, you can use a ruler to draw a straight line through them, extending it across the coordinate plane. This line represents all the points that satisfy the equation. Here's a rough step-by-step description of plotting the points and drawing the line: - Place a dot at (4, 2). - From (4, 2), move 5 units to the right to get to (9, 2). - From (9, 2), move 2 units down to get to (9, 0). - Place a dot at (9, 0). - Connect the two dots with a straight line. - Extend the line through and beyond both points to indicate that it continues infinitely in both directions. If you're graphing this by hand or using a graphing tool, make sure to label your axes and scale appropriately to accurately depict the slope and points.
Graphing a Line Using Point-Slope Form
To graph the line given by the equation \( y - 2 = \frac{-2}{5}(x - 4) \), you can follow these steps: 1. **Identify the slope and y-intercept:** The equation is in point-slope form, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. In this case, the slope (\( m \)) is \( \frac{-2}{5} \) and the point on the line is (4, 2). 2. **Plot the point on the graph:** Begin by plotting the point (4, 2) on your graph paper. This will be a point through which the line passes. 3. **Use the slope to find another point:** The slope \( \frac{-2}{5} \) means you move 2 units down and 5 units to the right from the point (4, 2) to find another point on the line. You could also move 2 units up and 5 units to the left if you wish to go in the opposite direction. 4. **Draw the line:** Connect the two points with a straight line. This line will extend infinitely in both directions. It's important to be precise and to plot points correctly. If you have graph paper, use a ruler to ensure that the line passes through the points accurately and extends across the grid.
Graphing Equations in Slope-Intercept Form
To graph the equation \( y + 6 = -\frac{3}{4}(x + 3) \), you should first rewrite it in slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Starting with the given equation: \( y + 6 = -\frac{3}{4}(x + 3) \) Isolate \( y \) on one side of the equation: \( y = -\frac{3}{4}(x + 3) - 6 \) Now distribute \( -\frac{3}{4} \) to both terms inside the parentheses: \( y = -\frac{3}{4}x - \frac{3}{4} \cdot 3 - 6 \) Multiply \( -\frac{3}{4} \) by 3 to get \( -\frac{9}{4} \), and then continue to simplify: \( y = -\frac{3}{4}x - \frac{9}{4} - 6 \) Since \( 6 = \frac{24}{4} \), we have: \( y = -\frac{3}{4}x - \frac{9}{4} - \frac{24}{4} \) Combine the constant terms: \( y = -\frac{3}{4}x - \frac{33}{4} \) So the equation in slope-intercept form is: \( y = -\frac{3}{4}x - \frac{33}{4} \) Now you can graph the line with the slope \( -\frac{3}{4} \) and a y-intercept \( -\frac{33}{4} \). Start by plotting the y-intercept on the y-axis at the point (0, -\( \frac{33}{4} \)) or (0, -8.25). Then, use the slope to find another point. From the y-intercept, move 3 units down and 4 units to the right (since the slope is negative), which will give you another point on the line. Connect these points with a straight line, and you will have graphed the given equation.
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