Example Question - y-intercept
Here are examples of questions we've helped users solve.
Explanation of Slope-Intercept Form
The given equation is the slope-intercept form of a line, which is represented as \( y = mx + b \), where: - \( m \) is the slope of the line - \( b \) is the y-intercept, i.e., the value of \( y \) when \( x = 0 \) There is no specific question to solve, as the image simply provides the general form of a linear equation in slope-intercept form.
Converting Function to Slope-Intercept Form
The function given in the image is: \[ f(x) = -2 \left( x + \frac{1}{4} \right) - \frac{49}{8} \] To express the function in the form of y-intercept, we need to simplify it while keeping it in the form of \( y = mx + b \), where \( m \) stands for the slope and \( b \) is the y-intercept. Let's simplify the function: \[ f(x) = -2x - \frac{1}{2} - \frac{49}{8} \] To combine the constant terms, we need a common denominator. The common denominator for \( \frac{1}{2} \) and \( \frac{49}{8} \) is 8. So let's convert \( \frac{1}{2} \) to have the denominator of 8: \[ - \frac{1}{2} = - \frac{4}{8} \] Now rewritting the equation we have: \[ f(x) = -2x - \frac{4}{8} - \frac{49}{8} = -2x - \frac{53}{8} \] So the function in the form of intercept is: \[ f(x) = -2x - \frac{53}{8} \] This function is now in slope-intercept form with \( m = -2 \) (the slope) and \( b = -\frac{53}{8} \) (the y-intercept).
Finding Zeros and Y-intercept of a Factored Function
The image displays a math problem related to finding the zeros of a function and the y-intercept of the function. The function is given in its factored form: f(x) = (x + 1)(x - 3)(x + 2) To find the zeros of the function, we need to set f(x) to 0 and solve for x. The zeros of a function are the x-values where the function crosses the x-axis (where f(x) = 0). Given that the function is already factored, the zeros are easily identified as the values that make each factor zero: x + 1 = 0 → x = -1 x - 3 = 0 → x = 3 x + 2 = 0 → x = -2 Thus, the zeros of the function are x = -1, 3, and -2. To find the y-intercept of the function, we need to find the value of f(x) when x = 0. The y-intercept is the point where the graph of the function crosses the y-axis (the value of f(0)). f(0) = (0 + 1)(0 - 3)(0 + 2) = (1)(-3)(2) = -6 Therefore, the y-intercept of the function is located at (0, -6). So, the correct answers to fill in the blanks would be: The zeros of the function f(x) = (x + 1)(x - 3)(x + 2) are -1, 3, and -2, and the y-intercept of the function is located at (0, -6).
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.
Writing Linear Equations in Slope-Intercept Form
Para escribir la ecuación de la línea en la forma pendiente-intersección, que es \( y = mx + b \), necesitamos encontrar dos piezas de información: la pendiente (m) y la intersección con el eje y (b). Observando la imagen, podemos identificar dos puntos por los cuales pasa la línea. En este caso, parece que podemos usar los puntos (0, 4) y (-2, 0). Si no fueran claros en la imagen, deberíamos buscar dos puntos donde la línea cruza exactamente a través de intersecciones de la cuadrícula para ser eso precisos. La pendiente (m) se calcula con la fórmula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Utilizando los puntos (0, 4) y (-2, 0): \[ m = \frac{0 - 4}{-2 - 0} = \frac{-4}{-2} = 2 \] Ahora que tenemos la pendiente, necesitamos la intersección con el eje y (b), que es simplemente el valor de y donde la línea cruza el eje y. Observando el punto que ya hemos identificado como (0, 4), podemos ver que b = 4. Por lo tanto, la ecuación de la línea en la forma pendiente-intersección es: \[ y = 2x + 4 \]
Finding the Y-Intercept of a Quadratic Equation
The image shows a math problem with the equation y = 3x^2 + 6x + 2 and the instructions to find the y-intercept of the equation. To find the y-intercept of a quadratic equation in the form of y = ax^2 + bx + c, we need to determine the value of y when x is equal to 0. So we will substitute x = 0 into the equation: y = 3(0)^2 + 6(0) + 2 y = 0 + 0 + 2 y = 2 Therefore, the y-intercept of the equation y = 3x^2 + 6x + 2 is 2.
Determining Equations of Lines and Inequalities
To find the equation of the line in part a, we need to determine the slope and the y-intercept. The equation of a line in slope-intercept form is: y = mx + b where m is the slope of the line, and b is the y-intercept. We can find the slope by looking at two points on the line and using the slope formula: slope (m) = (change in y) / (change in x) From the image, we can pick two points that the line passes through. Let's choose (0, -2) and (2, -1) since they're clearly on the grid intersections. Now, we find the slope: m = (y2 - y1) / (x2 - x1) m = (-1 - (-2)) / (2 - 0) m = (1) / (2) m = 1/2 Next, we find the y-intercept (b). This is where the line crosses the y-axis. Looking at the graph, we can see that this occurs at (0, -2), so b = -2. Now we have the slope m = 1/2 and y-intercept b = -2, the equation of the line is: y = (1/2)x - 2 To find the inequalities for parts b and c, we would normally look for instructions that designate whether the area above or below the line should be shaded. Since the image only shows the line and does not specify the inequalities or shaded regions, we can't determine what the inequalities would be. However, if you are being asked for the inequality that includes points below the line, the inequality would be: y ≤ (1/2)x - 2 If the inequality should include points above the line, it would be: y ≥ (1/2)x - 2 Remember, without additional information or context regarding which side of the line should be considered for the inequality, you cannot definitively state the inequality.
Calculating Slope and Y-Intercept from a Line Graph
To find the slope of the line represented in the graph, you need to pick two points on the line that you can clearly identify the coordinates for. Then, use the formula for slope, which is: \[ slope (m) = \frac{change \, in \, y}{change \, in \, x} = \frac{y_2 - y_1}{x_2 - x_1} \] Looking at the graph, let's pick the two points where the line intersects the grid lines. One point could be at \( (1, 1) \) and another point could be at \( (3, 7) \). Now using these points: \[ x_1 = 1, y_1 = 1 \] \[ x_2 = 3, y_2 = 7 \] Let's calculate the slope: \[ slope (m) = \frac{7 - 1}{3 - 1} = \frac{6}{2} = 3 \] Next, to find the y-intercept (the point where the line crosses the y-axis), look at where the line intersects the y-axis. From the graph, it appears that the line crosses the y-axis at \( (0, -1) \). Therefore, the y-intercept (b) is -1. So the slope is 3, and the y-intercept is -1.
Understanding a Line Equation in Slope-Intercept Form
It seems you've provided an image with an equation of a line: \( y = \frac{1}{2}x - 4 \). This equation is already in slope-intercept form, which is given by \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. From the given equation: - The slope (\( m \)) of the line is \( \frac{1}{2} \). - The y-intercept (\( b \)) of the line is \( -4 \). If you need to solve for a specific value, you would need to substitute that value for \( x \) and solve for \( y \), or vice versa. However, since there is no specific request on what to solve for, this is as much as can be said about the given equation. If you have a specific question about this equation, such as finding a point on the line or graphing the line, please provide further information.
Understanding Linear Equations: Slope-Intercept Form
The image you provided shows the equation y = 1/3x - 4. This is a linear equation in the slope-intercept form y = mx + b, where: - m stands for the slope of the line, which in this case is 1/3, indicating that for every increase of x by 3 units, y increases by 1 unit. - b stands for the y-intercept, which is the value of y when x is 0. In this case, b is -4, meaning the line crosses the y-axis at (0, -4). This equation does not have a specific solution since it represents a line with infinitely many points that satisfy the equation. To graph this line, you can plot the y-intercept at (0, -4) on a coordinate plane and use the slope to determine another point. Starting at (0, -4), move up 1 unit and to the right 3 units to find the next point at (3, -3). Connect these points with a straight line to represent the equation.
Quadratic Function Analysis
The image shows a question with two parts, (i) and (ii), concerning the graph of the function \(y = -3x^2 + 4x + 3\). (i) **Find the intercepts and the coordinates of the turning point on the graph of** \(y = -3x^2 + 4x + 3\). **To find the x-intercepts**, we set \(y\) equal to zero and solve for \(x\): \[0 = -3x^2 + 4x + 3\] This is a quadratic equation, and we can solve it using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -3\), \(b = 4\), and \(c = 3\). Let's solve for \(x\): \(x = \frac{-4 \pm \sqrt{4^2 - 4(-3)(3)}}{2(-3)}\) \(x = \frac{-4 \pm \sqrt{16 + 36}}{-6}\) \(x = \frac{-4 \pm \sqrt{52}}{-6}\) \(x = \frac{-4 \pm 2\sqrt{13}}{-6}\) Since we cannot simplify the square root any further, we have: \(x_1 = \frac{-4 + 2\sqrt{13}}{-6}\) and \(x_2 = \frac{-4 - 2\sqrt{13}}{-6}\) **To find the y-intercept**, we set \(x\) equal to zero: \(y = -3(0)^2 + 4(0) + 3 = 3\) So the y-intercept is at the point (0, 3). **To find the turning point** (also known as the vertex), we can use the formula for the x-coordinate of the vertex, \(x = -\frac{b}{2a}\), where \(a = -3\) and \(b = 4\): \(x = -\frac{4}{2(-3)}\) \(x = \frac{4}{6}\) \(x = \frac{2}{3}\) Now substitute \(x = \frac{2}{3}\) into the equation to find the y-coordinate: \(y = -3\left(\frac{2}{3}\right)^2 + 4\left(\frac{2}{3}\right) + 3\) \(y = -3\left(\frac{4}{9}\right) + \frac{8}{3} + 3\) \(y = -\frac{4}{3} + \frac{8}{3} + 3\) \(y = \frac{4}{3} + 3\) \(y = \frac{4}{3} + \frac{9}{3}\) \(y = \frac{13}{3}\) So the coordinates of the turning point are \( \left(\frac{2}{3}, \frac{13}{3}\right) \). (ii) **Use the graph to explain why** \(y = -3x^2 + 4x + 3\) **has two distinct real roots.** The question is asking to use the graph as an explanation, but since we cannot see the graph, I will provide a mathematical explanation based on what we have calculated. The quadratic equation \(y = -3x^2 + 4x + 3\) has two distinct real roots because the square root part of the quadratic formula resulted in \(\sqrt{52}\), which is a real number (as all square roots of positive numbers are real). Having two distinct roots means that the discriminant \(b^2 - 4ac\) is positive (\(52\) in our case), and this leads to two different x-intercepts on the graph, which we calculated earlier. In the context of a graph, this means that the parabola crosses the x-axis at two points, corresponding to the two distinct real roots.
Analysis of a Parabola in Vertex Form
The equation provided in the image is a parabola in vertex form: \[ y = 3(x - 1)^2 + 4 \] To find the coordinates of the turning point (vertex) of the parabola, we can directly read them from the vertex form of the quadratic equation: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In this equation, \(a = 3\), \(h = 1\), and \(k = 4\). Thus, the coordinates of the turning point (vertex) are: \[ (h, k) = (1, 4) \] To find the x-intercepts, set \(y = 0\) and solve for \(x\): \[ 0 = 3(x - 1)^2 + 4 \] \[ -4 = 3(x - 1)^2 \] \[ \frac{-4}{3} = (x - 1)^2 \] Since the square of a real number is always non-negative, the equation \((x - 1)^2 = -4/3\) has no real solutions. Thus, there are no x-intercepts for this parabola because it doesn’t cross the x-axis. To find the y-intercept, set \(x = 0\) and solve for \(y\): \[ y = 3(0 - 1)^2 + 4 \] \[ y = 3(1) + 4 \] \[ y = 3 + 4 \] \[ y = 7 \] So the y-intercept is at \(y = 7\) or at the point \((0, 7)\) on the graph. Lastly, the line of symmetry of a parabola is a vertical line that passes through the vertex. For the given equation, it is the line \(x = h\), which in this case is: \[ x = 1 \] This is the line of symmetry of the graph of the given quadratic function.
Equation of Rational Function with Given Conditions
To write an equation for a rational function that satisfies the given conditions, we can follow these steps: 1. Vertical asymptotes at \( x = 5 \) and \( x = -6 \) are given by factors in the denominator that become zero at these x-values. Thus, the denominator will be \( (x - 5)(x + 6) \). 2. X-intercepts at \( x = -5 \) and \( x = 1 \) indicate where the numerator of the function becomes zero. Thus, the numerator will be a product of factors that become zero at these x-values, which gives us \( (x + 5)(x - 1) \). 3. The y-intercept is the point where the function crosses the y-axis, which happens when \( x = 0 \). To get a y-intercept at 6, we need the constant term of the numerator (when \( x = 0 \)) to give us a y-value of 6. Since the denominator at \( x = 0 \) will be \( (0 - 5)(0 + 6) = -30 \), to obtain a y-intercept of 6, the numerator must evaluate to \( -30 * 6 \) when \( x = 0 \). Therefore, we can multiply the numerator by \( -6 \) to get the desired y-intercept. Putting it all together, the rational function is: \( y = \frac{-6(x + 5)(x - 1)}{(x - 5)(x + 6)} \) This function satisfies the given conditions: - The vertical asymptotes are at \( x = 5 \) and \( x = -6 \). - The x-intercepts are at \( x = -5 \) and \( x = 1 \). - The y-intercept is at \( y = 6 \) because when \( x = 0 \), \( y = \frac{-6(0 + 5)(0 - 1)}{(0 - 5)(0 + 6)} = \frac{30}{-30} = -1 \cdot -6 = 6 \).
Finding a Rational Function Given Vertical, X-intercepts, and Y-intercept
To write an equation for a rational function that meets the given requirements, you need to consider the vertical asymptotes, x-intercepts, and y-intercept provided. Vertical asymptotes occur where the denominator of the rational function is zero. Thus, for vertical asymptotes at \( x = -6 \) and \( x = 3 \), the factors in the denominator will be \( (x + 6) \) and \( (x - 3) \), since setting either of these equal to zero gives the respective values of \( x \) for the asymptotes. X-intercepts occur where the numerator is zero. Given the x-intercepts at \( x = -3 \) and \( x = -1 \), the factors in the numerator will be \( (x + 3) \) and \( (x + 1) \), since setting these factors equal to zero will yield the x-intercepts. Combining these, we get a function of the form: \[ f(x) = \frac{a(x + 3)(x + 1)}{(x + 6)(x - 3)} \] where \( a \) is a constant we need to determine using the y-intercept. The y-intercept is found by setting \( x = 0 \) in the function. It is given that the y-intercept is 10; thus, we solve for \( a \) by substituting \( x = 0 \) into the equation to get \( f(0) = 10 \): \[ 10 = \frac{a(0 + 3)(0 + 1)}{(0 + 6)(0 - 3)} \] \[ 10 = \frac{3a}{-18} \] Now solve for \( a \): \[ a = \frac{10 \times (-18)}{3} = -60 \] Insert \( a \) back into the original equation to get the complete function: \[ f(x) = \frac{-60(x + 3)(x + 1)}{(x + 6)(x - 3)} \]
Linear Equation Word Problem with Intercepts
The image contains a word problem about Maggie working two jobs and a linear equation derived from it, with questions asking to find the x-intercept and y-intercept for that equation. The given equation is: 15x + 14y = 630 Part (b) of the question asks to find the x-intercept and y-intercept for the equation. To find the x-intercept, we set y to 0 and solve for x: 15x + 14(0) = 630 15x = 630 x = 630 / 15 x = 42 So the x-intercept is (42, 0). To find the y-intercept, we set x to 0 and solve for y: 15(0) + 14y = 630 14y = 630 y = 630 / 14 y = 45 So the y-intercept is (0, 45). Filling in the blanks from the image: x-intercept: (42, 0) y-intercept: (0, 45)
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