Problem 37
Question
Find all intercepts for the graph of each quadratic function. $$f(x)=16-x^{2}$$
Step-by-Step Solution
Verified Answer
The intercepts are (0, 16), (4, 0), and (-4, 0).
1Step 1 - Find the y-intercept
To find the y-intercept of a function, substitute 0 for x in the equation and solve for f(x). For the function given: When \(x = 0\), \(f(x) = 16 - (0)^2 = 16\). So, the y-intercept is (0, 16).
2Step 2 - Find the x-intercepts
To find the x-intercepts of a function, set \(f(x) = 0\) and solve for x. For the function given: \(0 = 16 - x^2\). Rearranging, \(x^2 = 16\). Taking the square root of both sides, we get \(x = \pm\sqrt{16}\), which simplifies to \(x = 4\) and \(x = -4\). So, the x-intercepts are (4, 0) and (-4, 0).
3Step 3 - Summarize the intercepts
Summarize the intercepts found in the previous steps. The y-intercept is (0, 16), and the x-intercepts are (4, 0) and (-4, 0).
Key Concepts
finding interceptsquadratic equationsgraph analysis
finding intercepts
Intercepts are key points where a graph intersects the x-axis or y-axis.
They help us understand the behavior and position of a quadratic function.
For a quadratic function like \(f(x) = 16 - x^2\), we find intercepts by solving for specific values of x and f(x).
To find the **y-intercept**, substitute 0 for x in the equation.
In our example: \(x = 0\), then \(f(x) = 16 - (0)^2 = 16\).
So, the y-intercept is (0, 16).
For the **x-intercepts**, set \(f(x) = 0\) and solve for x.
In this case: \(0 = 16 - x^2\). Rearrange it to \(x^2 = 16\).
Taking the square root gives \(x = \pm \sqrt{16}\), which simplifies to \(\textstyle x = 4 \) and \(\textstyle x = -4 \).
So, the x-intercepts are (4, 0) and (-4, 0).
They help us understand the behavior and position of a quadratic function.
For a quadratic function like \(f(x) = 16 - x^2\), we find intercepts by solving for specific values of x and f(x).
To find the **y-intercept**, substitute 0 for x in the equation.
In our example: \(x = 0\), then \(f(x) = 16 - (0)^2 = 16\).
So, the y-intercept is (0, 16).
For the **x-intercepts**, set \(f(x) = 0\) and solve for x.
In this case: \(0 = 16 - x^2\). Rearrange it to \(x^2 = 16\).
Taking the square root gives \(x = \pm \sqrt{16}\), which simplifies to \(\textstyle x = 4 \) and \(\textstyle x = -4 \).
So, the x-intercepts are (4, 0) and (-4, 0).
quadratic equations
Quadratic equations take the form \(ax^2 + bx + c = 0 \). They produce a graph with a U-shape called a parabola.
These equations can have a maximum of two real solutions.
A key feature is the vertex, which is the highest or lowest point on the graph.
The direction of the parabola (opening upwards or downwards) depends on the sign of the leading coefficient (a).
If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
Intercepts are essential in understanding the parabola's position.
The y-intercept occurs where the graph crosses the y-axis, and the x-intercepts occur where the graph crosses the x-axis.
These intercepts are derived directly from the quadratic formula.
For example, for \(f(x) = 16 - x^2\), it is a downward opening parabola because our \(a\) is -1.
These equations can have a maximum of two real solutions.
A key feature is the vertex, which is the highest or lowest point on the graph.
The direction of the parabola (opening upwards or downwards) depends on the sign of the leading coefficient (a).
If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
Intercepts are essential in understanding the parabola's position.
The y-intercept occurs where the graph crosses the y-axis, and the x-intercepts occur where the graph crosses the x-axis.
These intercepts are derived directly from the quadratic formula.
For example, for \(f(x) = 16 - x^2\), it is a downward opening parabola because our \(a\) is -1.
graph analysis
Analyzing the graph of a quadratic function provides insights into its properties and behavior.
Start by identifying the intercepts, which provide key points of intersection with the axes.
Next, determine the vertex and the axis of symmetry.
The vertex can be found using the formula \(x = -\frac{b}{2a}\).
In our function \(f(x) = 16 - x^2\), there is no linear term (b=0), so the vertex is at x=0.
The vertex here is (0,16).
The axis of symmetry is the vertical line passing through the vertex.
Plot important points, including the vertex and intercepts, to sketch the graph.
Ensure you understand how the parabola opens (upwards or downwards).
For \(f(x) = 16 - x^2\), the parabola opens downward.
These steps give a comprehensive view of the quadratic function's behavior.
Start by identifying the intercepts, which provide key points of intersection with the axes.
Next, determine the vertex and the axis of symmetry.
The vertex can be found using the formula \(x = -\frac{b}{2a}\).
In our function \(f(x) = 16 - x^2\), there is no linear term (b=0), so the vertex is at x=0.
The vertex here is (0,16).
The axis of symmetry is the vertical line passing through the vertex.
Plot important points, including the vertex and intercepts, to sketch the graph.
Ensure you understand how the parabola opens (upwards or downwards).
For \(f(x) = 16 - x^2\), the parabola opens downward.
These steps give a comprehensive view of the quadratic function's behavior.
Other exercises in this chapter
Problem 36
Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$-x^{2}+3 x-4=0$$
View solution Problem 36
Find the vertex for the graph of each quadratic function. $$y=3 x^{2}-2 x+1$$
View solution Problem 38
Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$v^{2}=3 v+5$$
View solution Problem 38
Find all intercepts for the graph of each quadratic function. $$f(x)=x^{2}-9$$
View solution