For the quadratic equation y = 8 - 2x2, find the x-intercepts, axis of symmetry, turning point, and range. For the linear equation y = -2x, find the gradient.

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Asked on May 9, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-09T13:51:45.024Z

Another one Until — let's solve it. 1.1 Given y = 8 - 2x^2 x-intercepts: Step 1: Set y = 0 to find the x-intercepts. 0 = 8 - 2x^2 Step 2: Solve for x. 2x^2 = 8 x^2 = 4 x = ±sqrt(4) x = ± 2 The x-intercepts are (-2, 0) and (2, 0). Axis of symmetry: Step 1: For a quadratic function y = ax^2 + bx + c, the axis of symmetry is given by x = -(b)/(2a). In y = 8 - 2x^2, a = -2 and b = 0. x = -(0)/(2(-2)) x = 0 The axis of symmetry is x = 0. Turning point: Step 1: The x-coordinate of the turning point is the axis of symmetry, x = 0. Step 2: Substitute x = 0 into the equation to find the y-coordinate. y = 8 - 2(0)^2 y = 8 - 0 y = 8 The turning point is (0, 8). Range of y: Step 1: Since the coefficient of x^2 is negative (a = -2), the parabola opens downwards, meaning the turning point is a maximum point. Step 2: The maximum y-value is the y-coordinate of the turning point, which is 8. The range of y is y 8. 1.2 Below is the graph of y = -2x 1.2.1 Write down the gradient of the graph: Step 1: The equation of a straight line is y = mx + c, where m is the gradient. In y = -2x, m = -2. The gradient is -2. 1.2.2 What is the mathematical name of the graph? The graph of y = -2x is a linear function (or straight line). 1.3 Given y = 3^x 1.3.1 Tabulate values of x for -2 x 2, x an integer and calculate the corresponding y-values if y = 3^x. Step 1: Identify integer values for x in the given range: x = -2, -1, 0, 1, 2. Step 2: Calculate the corresponding y values using y = 3^x. • For x = -2, y = 3^-2 = (1)/(3^2) = (1)/(9). • For x = -1, y = 3^-1 = (1)/(3). • For x = 0, y = 3^0 = 1. • For x = 1, y = 3^1 = 3. • For x = 2, y = 3^2 = 9. The tabulated values are: |c|c| x & y \\ -2 & (1)/(9) \\ -1 & (1)/(3) \\ 0 & 1 \\ 1 & 3 \\ 2 & 9 \\ 1.3.2 Use the values in question 1.3.1 to sketch the graph of y = 3^x on the ADDENDUM provided. To sketch the graph, plot the points (-2, (1)/(9)), (-1, (1)/(3)), (0, 1), (1, 3), and (2, 9) on a coordinate plane and connect them with a smooth curve. 1.3.3 Is the graph drawn in question 1.3.2 continuous or discontinuous? The graph of y = 3^x is continuous. 1.3.4 Write down the new equation of y if the graph moves two units down. Step 1: Moving a graph down by k units means subtracting k from the original equation. Step 2: The original equation is y = 3^x. Moving it two units down gives y = 3^x - 2. The new equation is y = 3^x - 2. Send me the next one 📸

Accepted Answer · 2026-05-09T13:51:45.024Z

Another one Until — let's solve it. 1.1 Given y = 8 - 2x^2 x-intercepts: Step 1: Set y = 0 to find the x-intercepts. 0 = 8 - 2x^2 Step 2: Solve for x. 2x^2 = 8 x^2 = 4 x = ±sqrt(4) x = ± 2 The x-intercepts are (-2, 0) and (2, 0). Axis of symmetry: Step 1: For a quadratic function y = ax^2 + bx + c, the axis of symmetry is given by x = -(b)/(2a). In y = 8 - 2x^2, a = -2 and b = 0. x = -(0)/(2(-2)) x = 0 The axis of symmetry is x = 0. Turning point: Step 1: The x-coordinate of the turning point is the axis of symmetry, x = 0. Step 2: Substitute x = 0 into the equation to find the y-coordinate. y = 8 - 2(0)^2 y = 8 - 0 y = 8 The turning point is (0, 8). Range of y: Step 1: Since the coefficient of x^2 is negative (a = -2), the parabola opens downwards, meaning the turning point is a maximum point. Step 2: The maximum y-value is the y-coordinate of the turning point, which is 8. The range of y is y 8. 1.2 Below is the graph of y = -2x 1.2.1 Write down the gradient of the graph: Step 1: The equation of a straight line is y = mx + c, where m is the gradient. In y = -2x, m = -2. The gradient is -2. 1.2.2 What is the mathematical name of the graph? The graph of y = -2x is a linear function (or straight line). 1.3 Given y = 3^x 1.3.1 Tabulate values of x for -2 x 2, x an integer and calculate the corresponding y-values if y = 3^x. Step 1: Identify integer values for x in the given range: x = -2, -1, 0, 1, 2. Step 2: Calculate the corresponding y values using y = 3^x. • For x = -2, y = 3^-2 = (1)/(3^2) = (1)/(9). • For x = -1, y = 3^-1 = (1)/(3). • For x = 0, y = 3^0 = 1. • For x = 1, y = 3^1 = 3. • For x = 2, y = 3^2 = 9. The tabulated values are: |c|c| x & y \\ -2 & (1)/(9) \\ -1 & (1)/(3) \\ 0 & 1 \\ 1 & 3 \\ 2 & 9 \\ 1.3.2 Use the values in question 1.3.1 to sketch the graph of y = 3^x on the ADDENDUM provided. To sketch the graph, plot the points (-2, (1)/(9)), (-1, (1)/(3)), (0, 1), (1, 3), and (2, 9) on a coordinate plane and connect them with a smooth curve. 1.3.3 Is the graph drawn in question 1.3.2 continuous or discontinuous? The graph of y = 3^x is continuous. 1.3.4 Write down the new equation of y if the graph moves two units down. Step 1: Moving a graph down by k units means subtracting k from the original equation. Step 2: The original equation is y = 3^x. Moving it two units down gives y = 3^x - 2. The new equation is y = 3^x - 2. Send me the next one 📸

For the quadratic equation y = 8 - 2x2, find the x-intercepts, axis of symmetry, turning point, and range. For the linear equation y = -2x, find the gradient.

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For the quadratic equation y = 8 - 2x2, find the x-intercepts, axis of symmetry, turning point, and range. For the linear equation y = -2x, find the gradient.

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