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Chapter 10: Problem 61

Find the \(x\) - and \(y\) -intercepts of the graph of the quadratic function. See Example \(9 .\) $$ f(x)=-2 x^{2}+4 x $$

Short Answer

Expert verified
The x-intercepts are (0, 0) and (2, 0), and the y-intercept is (0, 0).

Step by step solution

01

Identify the Quadratic Function

The given quadratic function is \( f(x) = -2x^2 + 4x \). We need to find both the \(x\)-intercepts and the \(y\)-intercept of this function.
02

Find the Y-intercept

The \(y\)-intercept occurs where \(x = 0\). Evaluate the function at \(x = 0\): \[ f(0) = -2(0)^2 + 4(0) = 0. \] Thus, the \(y\)-intercept is the point \((0, 0)\).
03

Find the X-intercepts

The \(x\)-intercepts occur where \(f(x) = 0\). Set the equation \(-2x^2 + 4x = 0\) and solve for \(x\): \[ -2x^2 + 4x = 0 \] Factor the expression: \[ -2x(x - 2) = 0. \] Set each factor equal to zero: \(-2x = 0\) and \(x - 2 = 0\). Solve: \[ x = 0 \text{ and } x = 2. \] Thus, the \(x\)-intercepts are at \((0, 0)\) and \((2, 0)\).
04

Conclusion

The \(x\)-intercepts are \((0, 0)\) and \((2, 0)\), and the \(y\)-intercept is \((0, 0)\). Therefore, both intercepts confirm that the point \((0, 0)\) serves as both an \(x\)- and \(y\)-intercept for this function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding X-Intercepts of Quadratic Functions
The x-intercepts of a quadratic function are the points where the graph of the function meets the x-axis. At these points, the value of the function is zero, meaning \(f(x) = 0\). In our case, the quadratic function given is \(f(x) = -2x^2 + 4x\). To find the x-intercepts, we need to solve the equation \(-2x^2 + 4x = 0\).

Here's a simple way to find them:
  • Factor the quadratic expression: \(-2x(x - 2) = 0\).
  • Set each factor to zero: \(-2x = 0\) and \(x - 2 = 0\).
  • Solve these simple equations to find the x-values: \(x = 0\) and \(x = 2\).
These solutions tell us that the graph crosses the x-axis at points \((0, 0)\) and \((2, 0)\). This means the parabola associated with this quadratic function has x-intercepts at these two points.
Grasping Y-Intercepts in Quadratic Functions
The y-intercept is where the graph of a function crosses the y-axis. This occurs where the input, or \(x\)-value, is zero. Finding the y-intercept is a matter of plugging \(x = 0\) into the quadratic function and solving for \(f(x)\). For the function \(f(x) = -2x^2 + 4x\), here鈥檚 how you locate it:
  • Substitute \(x = 0\) into the function: \(f(0) = -2(0)^2 + 4(0) = 0\).
Thus, the y-intercept of this function is \((0, 0)\). It is the same point where the graph intersects the x-axis, making it both an x-intercept and a y-intercept. This understanding helps identify that the graph of the quadratic touches the origin.
Basics of Quadratic Equations
Quadratic equations are a type of polynomial equation, involving an \(x^2\) term as its highest degree component. They take the standard form \(ax^2 + bx + c = 0\). In mathematics, these equations describe parabolas, or curved paths, when graphed.

Key features of quadratic functions like \(f(x) = -2x^2 + 4x\) often include:
  • The vertex, which is the peak or lowest point depending on the parabola's direction.
  • The axis of symmetry, a vertical line that divides the parabola into two mirror images.
  • Notable intercepts鈥攖he points where the function meets the axes of a graph: x-intercepts and the y-intercept.
Understanding quadratic equations also involves recognizing that if \(a\) is less than zero, as in our function, the parabola opens downwards. This implies that the vertex is a maximum point. Knowing these aspects can give insights into the behavior and properties of the quadratic function when represented graphically.

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Most popular questions from this chapter

Determine the coordinates of the vertex of the graph of each function using the vertex formula. Then determine the \(x-\) and \(y\) -intercepts of the graph. Finally, plot several points and complete the graph. See Example \(9 .\) $$ f(x)=-2 x^{2}+8 x-10 $$

Solve each inequality. Write the solution set in interval notation and graph it. $$ x^{2}-6 x+9<0 $$

Use completing the square to solve each equation. Approximate each solution to the nearest hundredth. See Example 7. $$ t^{2}+16 t-16=0 $$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ 5 x^{2}+40=0 $$

Simplify each expression. Assume all variables represent positive numbers. $$ \frac{\sqrt{3}}{\sqrt{50}} $$
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