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Magazine Chapter 11: Problem 26
Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and graph the function. $$ f(x)=3 x^{2}+12 x+16 $$
Short Answer
Expert verified
The vertex is (-2, 4) and the parabola opens upward.
Step by step solution
01
Identify the Quadratic Function
The given quadratic function is \( f(x) = 3x^2 + 12x + 16 \). This is in the standard form \( ax^2 + bx + c \), where \( a = 3 \), \( b = 12 \), and \( c = 16 \).
02
Determine If the Parabola Opens Upward or Downward
To determine whether the parabola opens upward or downward, look at the coefficient \( a \). Since \( a = 3 \) is positive, the parabola opens upward.
03
Find the Vertex
The vertex \((h, k)\) of a parabola given by \( ax^2 + bx + c \) can be found using the formula \( h = -\frac{b}{2a} \). Substitute \( b = 12 \) and \( a = 3 \):\[h = -\frac{12}{2 \times 3} = -2\]Plug \( h = -2 \) into the function to find \( k \):\[k = f(-2) = 3(-2)^2 + 12(-2) + 16 = 12 - 24 + 16 = 4\]Thus, the vertex is \((-2, 4)\).
04
Find the Intercepts
**Y-intercept:** The y-intercept is found by evaluating \( f(0) \):\[f(0) = 3(0)^2 + 12(0) + 16 = 16\]**X-intercepts:** Solve \( 3x^2 + 12x + 16 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):\[x = \frac{-12 \pm \sqrt{12^2 - 4 \times 3 \times 16}}{6}\]\[x = \frac{-12 \pm \sqrt{144 - 192}}{6} = \frac{-12 \pm \sqrt{-48}}{6}\]Since the discriminant \(-48\) is negative, there are no real x-intercepts.
05
Sketch the Graph
To graph the function, plot the vertex \((-2, 4)\) and y-intercept \((0, 16)\). Since the parabola opens upward and there are no real x-intercepts, the graph is a U-shaped curve passing through these points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabolas
A parabola is the graph of a quadratic function. You can recognize these by their characteristic U-shape.
Parabolas can open either upward or downward, depending on the coefficient of the squared term in the quadratic equation.
Understanding a parabola is crucial when dealing with quadratic functions because it tells us a lot about how the function behaves.
Parabolas can open either upward or downward, depending on the coefficient of the squared term in the quadratic equation.
Understanding a parabola is crucial when dealing with quadratic functions because it tells us a lot about how the function behaves.
- If the coefficient ("a") of the quadratic term is positive, the parabola opens upward, resembling a smiley face.
- If this coefficient is negative, the parabola opens downward, resembling a frown.
- A parabola is symmetrical around its vertex, making the vertex a key point of reference.
Vertex of a Parabola
The vertex of a parabola is the highest or lowest point on its graph, depending on whether the parabola opens downward or upward.
For a quadratic function in standard form, given by \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula for the x-coordinate \( h = -\frac{b}{2a} \).
Plugging this value back into the function gives us the y-coordinate of the vertex.
For a quadratic function in standard form, given by \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula for the x-coordinate \( h = -\frac{b}{2a} \).
Plugging this value back into the function gives us the y-coordinate of the vertex.
- The vertex formula, \( h = -\frac{b}{2a} \), helps in determining the axis of symmetry of the parabola.
- The x-value of the vertex is where this symmetry line lies.
- In our example, the vertex \((-2, 4)\) is where the parabola changes direction.
X-Intercepts
X-intercepts are the points where the graph of the function crosses the x-axis.
These points are found by solving the equation \( ax^2 + bx + c = 0 \) using the quadratic formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
These points are found by solving the equation \( ax^2 + bx + c = 0 \) using the quadratic formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- The discriminant \((b^2 - 4ac)\) within the quadratic formula determines the nature of the x-intercepts.
- If the discriminant is positive, there are two distinct real x-intercepts.
- If it is zero, there is exactly one real x-intercept (the vertex lies on the x-axis).
- If it is negative, as in our example, there are no real x-intercepts, indicating no crossing of the x-axis by the graph.
Y-Intercept
The y-intercept is where the graph crosses the y-axis. This occurs when \( x = 0 \).
You can find the y-intercept simply by substituting \( x = 0 \) into the quadratic function.
In our function \( f(x) = 3x^2 + 12x + 16 \), by plugging in 0, we find the y-intercept is 16.
You can find the y-intercept simply by substituting \( x = 0 \) into the quadratic function.
In our function \( f(x) = 3x^2 + 12x + 16 \), by plugging in 0, we find the y-intercept is 16.
- The y-intercept is an easy point to calculate since it involves no solving of equations.
- It provides a starting point for graphing, as it is straightforward to plot \( y = c \).
- Knowing the y-intercept helps to visualize the function's initial value when starting from \( x = 0 \).
