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Chapter 3: Problem 60

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((5,6) ;\) through \((1,-6)\)

Short Answer

Expert verified
The quadratic equation is \(P(x) = -\frac{3}{4}x^2 + \frac{15}{2}x - \frac{51}{4}\).

Step by step solution

01

Understand the Vertex Form

The vertex form of a quadratic function is \( P(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. From the provided data, we know the vertex is \((5, 6)\), so \(h = 5\) and \(k = 6\).
02

Substitute Vertex into Vertex Form

Substitute \(h = 5\) and \(k = 6\) into the vertex form. The equation becomes: \[ P(x) = a(x-5)^2 + 6 \]
03

Use the Additional Point to Find \(a\)

The quadratic passes through the point \((1, -6)\). Substitute \(x = 1\) and \(P(x) = -6\) into the equation to solve for \(a\):\[ -6 = a(1-5)^2 + 6 \]\[ -6 = a(16) + 6 \]Solving for \(a\) gives: \[ -12 = 16a \]\[ a = -\frac{12}{16} = -\frac{3}{4} \]
04

Construct the Equation in Vertex Form

Now that \(a = -\frac{3}{4}\), substitute it back into the vertex form: \[ P(x) = -\frac{3}{4}(x-5)^2 + 6 \]
05

Expand to Standard Form

Expand \((x-5)^2\) and substitute back:\[ P(x) = -\frac{3}{4}((x^2 - 10x + 25)) + 6 \] Distribute \(-\frac{3}{4}\): \[ = -\frac{3}{4}x^2 + \frac{30}{4}x - \frac{75}{4} + 6 \] Combine constants: Convert 6 to \(\frac{24}{4}\): \[ -\frac{75}{4} + \frac{24}{4} = -\frac{51}{4} \]Hence, the standard form is: \[ P(x) = -\frac{3}{4}x^2 + \frac{30}{4}x - \frac{51}{4} \]
06

Simplify Coefficients

Simplify the coefficients for clarity:\[ P(x) = -\frac{3}{4}x^2 + \frac{15}{2}x - \frac{51}{4} \]

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Key Concepts

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

Vertex Form
The vertex form of a quadratic function captures the essence of the parabola's shape by focusing on its vertex. The equation is given by \( P(x) = a(x-h)^2 + k \).
Here:
  • \(a\) controls the width and direction of the parabola.
  • \(h\) and \(k\) denote the vertex's position on the coordinate plane.
The vertex form is particularly advantageous when you know the vertex and desire to express the quadratic in a way that highlights this point. For instance, if the vertex is \((5, 6)\), as in the original exercise, we simply substitute \(h = 5\) and \(k = 6\) into the equation, resulting in \( P(x) = a(x-5)^2 + 6 \).
This form is very useful for finding a quadratic equation when given the vertex and another point on the parabola.
Standard Form
Switching a quadratic equation to the standard form can offer a more straightforward look at the coefficients that influence its graph. The standard form is: \( P(x) = ax^2 + bx + c \).

Key points of standard form are:
  • \(a\) dictates the parabola鈥檚 direction; a positive \(a\) opens upwards and a negative \(a\) opens downwards.
  • \(b\) affects the tilt of the parabola.
  • \(c\) represents the y-intercept, the point where the parabola crosses the y-axis.
Converting from vertex form to standard form often involves expanding and simplifying the equation. To do this:
  • Multiply out the squared term in \((x-h)^2\).
  • Distribute the \(a\) value.
  • Combine like terms.
For example, the vertex form \(-\frac{3}{4}(x-5)^2 + 6\) can be expanded and simplified to the standard form \(-\frac{3}{4}x^2 + \frac{15}{2}x - \frac{51}{4}\).
This form is especially beneficial for analyzing the effects of changes in \(a\), \(b\), and \(c\) on the position and shape of the parabola.
Parabola
A parabola is a U-shaped curve which can open upwards or downwards, dictated by the sign of the \(a\) coefficient in the quadratic function.
It is symmetric around a vertical line called the axis of symmetry, which passes through the vertex.Understanding parabolas starts with recognizing their essential features:
  • The vertex: The highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards.
  • The axis of symmetry: A vertical line that splits the parabola into two mirror-image halves.
The orientation and steepness of a parabola are directly influenced by \(a\), while its vertex's coordinates \((h, k)\), described in the vertex form, pinpoint its location. Parabolas appear in various real-life phenomena, such as the trajectories of projectiles or satellite dishes' shapes.
Quadratic Equation
A quadratic equation is a fundamental form of a polynomial equation of degree two.
The general structure is noted as \( ax^2 + bx + c = 0 \), where \(a eq 0\).

Key aspects of quadratic equations include:
  • The coefficient \(a\) must be non-zero to preserve the equation's quadratic nature.
  • The solutions or roots of the quadratic, given by the values of \(x\) that satisfy the equation.
Quadratic equations can be approached through:
  • Factoring the quadratic if possible.
  • Using the Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Completing the square, which involves manipulating the equation to derive the vertex form.
Quadratic equations are pivotal in numerous mathematical contexts, from calculating areas to modeling situations in physics and engineering. They demonstrate how changing a single coefficient can drastically alter solutions and graphs, enhancing our understanding of algebraic expressions.

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Most popular questions from this chapter

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