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Magazine Chapter 3: Problem 89
Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a>0, b^{2}-4 a c>0$$
Short Answer
Expert verified
The graph is an upward-opening parabola with two distinct real roots.
Step by step solution
01
Identify the shape of the parabola
The function provided is a quadratic function in the form of a parabola, given by \( f(x) = ax^2 + bx + c \). The value of \(a\) is greater than 0 (i.e., \(a > 0\)), which implies that the parabola opens upwards.
02
Determine the discriminant
For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \(\Delta\) is calculated as \( b^2 - 4ac \). Here, it is given that \( b^2 - 4ac > 0 \), indicating that the parabola has two distinct real roots.
03
Analyze the vertex and axis of symmetry
The vertex of a parabola \( ax^2 + bx + c \) is at the point \( x = -\frac{b}{2a} \). Since \( a > 0 \), the vertex will be the minimum point. The axis of symmetry is a vertical line: \( x = -\frac{b}{2a} \).
04
Sketch the graph
Begin by plotting the vertex at \( x = -\frac{b}{2a} \) on the graph. Since \( b^2 - 4ac > 0 \), there will be two x-intercepts symmetrically placed around the vertex. Since the parabola opens upwards, draw a U-shaped curve passing through the x-intercepts and vertex.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic function, commonly described by the equation \(f(x) = ax^2 + bx + c\). The sign of \(a\) determines the direction in which the parabola opens. If \(a > 0\), the parabola opens upwards, resembling a smile. On the other hand, if \(a < 0\), it opens downwards, which looks more like a frown.
Parabolas have various real-world applications such as in physics where they describe the trajectories of projectiles or reflectors in satellite dishes and headlights.
Some important properties of parabolas include:
Parabolas have various real-world applications such as in physics where they describe the trajectories of projectiles or reflectors in satellite dishes and headlights.
Some important properties of parabolas include:
- The vertex as the point of turning.
- An axis of symmetry that vertically divides it into two equal halves.
- Zero to two real roots, depending on parameters, indicated by the discriminant.
Discriminant
The discriminant is a key concept in understanding quadratic functions. It provides crucial information about the roots of a quadratic equation \(ax^2 + bx + c = 0\). The discriminant, denoted by \(\Delta\), is calculated as \(b^2 - 4ac\).
Here's what the discriminant can tell us:
Here's what the discriminant can tell us:
- If \(\Delta > 0\), there are two distinct real roots. The parabola will cross the x-axis at two different points.
- If \(\Delta = 0\), there is exactly one real root, meaning the parabola just touches the x-axis. This is also known as a repeated or double root.
- If \(\Delta < 0\), there are no real roots. The parabola does not intersect the x-axis at any real point.
Vertex
The vertex of a parabola is its most significant point. For a function \(f(x) = ax^2 + bx + c\), the vertex provides information about the maximum or minimum value of the graph. You can find the x-coordinate of the vertex using the formula \(-\frac{b}{2a}\). The corresponding y-coordinate is found by substituting this x-value into the function.
The vertex is:
The vertex is:
- A minimum point if \(a > 0\), because the parabola opens upwards.
- A maximum point if \(a < 0\), because the parabola opens downwards.
Axis of Symmetry
The axis of symmetry is an essential aspect of parabolas. It is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. This line allows for both sides of the parabola to be symmetrical.
For a quadratic function \(f(x) = ax^2 + bx + c\), the formula for the axis of symmetry is given by \(x = -\frac{b}{2a}\). This equation gives a consistent method to identify the line of symmetry regardless of the specific parameters in the quadratic function.
Knowing the axis of symmetry helps with:
For a quadratic function \(f(x) = ax^2 + bx + c\), the formula for the axis of symmetry is given by \(x = -\frac{b}{2a}\). This equation gives a consistent method to identify the line of symmetry regardless of the specific parameters in the quadratic function.
Knowing the axis of symmetry helps with:
- Identifying the parabola's midpoint.
- Ensuring accuracy in graph sketches.
- Understanding how changes to the equation affect the graph's shape.
